https://lambdia.com https://lambdia.com/fr https://lambdia.com/es https://lambdia.com/start https://lambdia.com/fr/start https://lambdia.com/es/start https://lambdia.com/quant-interview https://lambdia.com/fr/quant-interview https://lambdia.com/es/quant-interview https://lambdia.com/quant-interview/derivatives https://lambdia.com/fr/quant-interview/derivatives https://lambdia.com/es/quant-interview/derivatives https://lambdia.com/quant-interview/brainteasers https://lambdia.com/fr/quant-interview/brainteasers https://lambdia.com/es/quant-interview/brainteasers https://lambdia.com/quant-interview/probability https://lambdia.com/fr/quant-interview/probability https://lambdia.com/es/quant-interview/probability https://lambdia.com/quant-interview/market-making https://lambdia.com/fr/quant-interview/market-making https://lambdia.com/es/quant-interview/market-making https://lambdia.com/quant-interview/mental-math https://lambdia.com/fr/quant-interview/mental-math 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https://lambdia.com/es/guides/quant-internship-timeline https://lambdia.com/firms https://lambdia.com/fr/firms https://lambdia.com/es/firms https://lambdia.com/firms/jane-street https://lambdia.com/fr/firms/jane-street https://lambdia.com/es/firms/jane-street https://lambdia.com/firms/optiver https://lambdia.com/fr/firms/optiver https://lambdia.com/es/firms/optiver https://lambdia.com/firms/citadel-securities https://lambdia.com/fr/firms/citadel-securities https://lambdia.com/es/firms/citadel-securities https://lambdia.com/firms/hudson-river-trading https://lambdia.com/fr/firms/hudson-river-trading https://lambdia.com/es/firms/hudson-river-trading https://lambdia.com/firms/jump-trading https://lambdia.com/fr/firms/jump-trading https://lambdia.com/es/firms/jump-trading https://lambdia.com/firms/imc https://lambdia.com/fr/firms/imc https://lambdia.com/es/firms/imc https://lambdia.com/firms/flow-traders https://lambdia.com/fr/firms/flow-traders https://lambdia.com/es/firms/flow-traders https://lambdia.com/firms/sig https://lambdia.com/fr/firms/sig https://lambdia.com/es/firms/sig https://lambdia.com/firms/two-sigma https://lambdia.com/fr/firms/two-sigma https://lambdia.com/es/firms/two-sigma https://lambdia.com/firms/de-shaw https://lambdia.com/fr/firms/de-shaw https://lambdia.com/es/firms/de-shaw https://lambdia.com/math https://lambdia.com/fr/math https://lambdia.com/es/math https://lambdia.com/math/foundations-logic https://lambdia.com/fr/math/foundations-logic https://lambdia.com/es/math/foundations-logic https://lambdia.com/math/number-theory https://lambdia.com/fr/math/number-theory https://lambdia.com/es/math/number-theory https://lambdia.com/math/algebra https://lambdia.com/fr/math/algebra https://lambdia.com/es/math/algebra https://lambdia.com/math/combinatorics-discrete https://lambdia.com/fr/math/combinatorics-discrete https://lambdia.com/es/math/combinatorics-discrete https://lambdia.com/math/geometry https://lambdia.com/fr/math/geometry https://lambdia.com/es/math/geometry https://lambdia.com/math/topology https://lambdia.com/fr/math/topology https://lambdia.com/es/math/topology https://lambdia.com/math/analysis-calculus https://lambdia.com/fr/math/analysis-calculus https://lambdia.com/es/math/analysis-calculus https://lambdia.com/math/diffeq-dynamics https://lambdia.com/fr/math/diffeq-dynamics https://lambdia.com/es/math/diffeq-dynamics https://lambdia.com/math/probability-statistics https://lambdia.com/fr/math/probability-statistics https://lambdia.com/es/math/probability-statistics https://lambdia.com/math/numerical-computational https://lambdia.com/fr/math/numerical-computational https://lambdia.com/es/math/numerical-computational https://lambdia.com/math/applied-interdisciplinary https://lambdia.com/fr/math/applied-interdisciplinary https://lambdia.com/es/math/applied-interdisciplinary https://lambdia.com/math/math-computer-science https://lambdia.com/fr/math/math-computer-science https://lambdia.com/es/math/math-computer-science https://lambdia.com/math/elementary-school https://lambdia.com/fr/math/elementary-school https://lambdia.com/es/math/elementary-school https://lambdia.com/math/recreational-history-meta https://lambdia.com/fr/math/recreational-history-meta https://lambdia.com/es/math/recreational-history-meta https://lambdia.com/search https://lambdia.com/fr/search https://lambdia.com/es/search https://lambdia.com/tos https://lambdia.com/fr/tos https://lambdia.com/privacy https://lambdia.com/fr/privacy https://lambdia.com/problems/correlation-transitivity 2026-08-17T09:00:00.000Z https://lambdia.com/problems/kelly-criterion 2026-08-19T06:00:00.000Z https://lambdia.com/problems/broken-stick 2026-08-09T00:00:00.000Z https://lambdia.com/problems/birthday-problem 2026-08-08T06:00:00.000Z https://lambdia.com/problems/hundred-prisoners 2026-08-07T06:00:00.000Z https://lambdia.com/problems/monty-hall 2026-08-06T06:00:00.000Z https://lambdia.com/problems/twelve-coins 2026-08-19T18:00:00.000Z https://lambdia.com/problems/two-envelopes 2026-08-22T15:00:00.000Z https://lambdia.com/problems/winners-curse 2026-08-22T18:00:00.000Z https://lambdia.com/problems/water-jug 2026-08-10T06:00:00.000Z https://lambdia.com/problems/counterfeit-bag 2026-08-10T09:00:00.000Z https://lambdia.com/problems/burning-ropes 2026-08-10T12:00:00.000Z https://lambdia.com/problems/gold-bar 2026-08-10T15:00:00.000Z https://lambdia.com/problems/towers-of-hanoi 2026-08-10T18:00:00.000Z https://lambdia.com/problems/spider-and-fly 2026-08-11T12:00:00.000Z https://lambdia.com/problems/water-and-wine 2026-08-11T15:00:00.000Z https://lambdia.com/problems/three-switches 2026-08-13T15:00:00.000Z https://lambdia.com/problems/hat-puzzle 2026-08-13T18:00:00.000Z https://lambdia.com/problems/fly-between-trains 2026-08-07T12:00:00.000Z https://lambdia.com/problems/jeep-problem 2026-08-08T06:00:00.000Z https://lambdia.com/problems/truel 2026-08-09T12:00:00.000Z https://lambdia.com/problems/mutilated-chessboard 2026-08-09T15:00:00.000Z https://lambdia.com/problems/walk-south-east-north 2026-08-09T18:00:00.000Z https://lambdia.com/problems/snail-on-a-pole 2026-08-03T08:00:00.000Z https://lambdia.com/problems/russian-roulette 2026-08-02T11:00:00.000Z https://lambdia.com/problems/false-positive-paradox 2026-07-31T00:00:00.000Z https://lambdia.com/problems/coin-weighing 2026-08-01T01:00:00.000Z https://lambdia.com/problems/boy-girl 2026-08-05T06:00:00.000Z https://lambdia.com/a/seventy-percent-each-and-still-negative Both stocks track the index at 0.7. What about each other? https://i.ytimg.com/vi/8CL2LKgk_ls/hqdefault.jpg The two correlations you are handed do not pin the third one down, but they fence it into exactly [−1/50, 1], and that interval dips below zero. The fence falls out of a 3×3 determinant read as a quadratic in the unknown, and out of a picture: 0.7 is an angle of 45.573°, both stocks live on a cone of that half-angle around the index, and putting them on opposite sides opens 91.146° between them. Also here: why the real tipping point is ab ≥ 0 together with a² + b² ≥ 1 rather than "both above 0.707", why 0.9 and 0.5 force a positive answer while 0.9 and −0.9 allow −1, why standing on the floor costs a rank, and why three Bernoulli(0.5) indicators with the same two correlations are confined to [0.40, 1] instead. https://www.youtube-nocookie.com/embed/8CL2LKgk_ls 2026-08-17T09:00:00Z yes Lambdia 2026-08-17T09:00:00.000Z https://lambdia.com/a/double-the-right-bet-and-you-go-backwards A coin lands heads 60% of the time. How much do you bet? https://i.ytimg.com/vi/R36uzfgZYK8/hqdefault.jpg The fraction that maximises long-run growth is exactly the edge, 2p-1, which is 0.2 on this coin, and one derivative gets you there. Double it and the growth rate is -0.0024469 a flip, negative on a game that leans your way three hundred times in a row, and the crossing happens at 0.3894 rather than at 0.4. The article carries the exact median over 300 flips, 25 dollars to 10504.19 at the optimum and to 12.00 at double, the reason about 48 percent of overbettors still finish ahead anyway, and the place where the textbook approximation mean minus half the variance returns the opposite sign. https://www.youtube-nocookie.com/embed/R36uzfgZYK8 2026-08-19T06:00:00Z yes Lambdia 2026-08-19T06:00:00.000Z https://lambdia.com/a/broken-stick-triangle Snap a stick twice. Do the pieces close into a triangle? https://i.ytimg.com/vi/ILkN950o-tY/hqdefault.jpg Two random breaks, three pieces, and three inequalities that collapse into one. The quarter falls out of a square with no integral at all, and the average longest piece, 11/18, explains why the answer feels too low but isn't. https://www.youtube-nocookie.com/embed/ILkN950o-tY 747 2026-08-09 yes Lambdia 2026-08-09T00:00:00.000Z https://lambdia.com/a/twenty-three-people-share-a-birthday 2026-08-08T06:00:00.000Z https://lambdia.com/a/hundred-prisoners-survive-31-percent 2026-08-07T06:00:00.000Z https://lambdia.com/a/switching-doors-wins-two-thirds 2026-08-06T06:00:00.000Z https://lambdia.com/a/variance-that-does-not-scale-kills-the-random-walk 2026-08-31T18:00:00.000Z https://lambdia.com/a/a-call-has-six-times-the-beta 2026-09-01T06:00:00.000Z https://lambdia.com/a/gamma-beats-time-decay-overnight 2026-09-01T09:00:00.000Z https://lambdia.com/a/mean-reversion-makes-options-cheaper 2026-09-01T12:00:00.000Z https://lambdia.com/a/no-arbitrage-erases-risk-preferences 2026-09-01T15:00:00.000Z https://lambdia.com/a/swap-curve-discounting-lifts-a-bond-above-par 2026-09-01T18:00:00.000Z https://lambdia.com/a/mortgage-bonds-have-negative-convexity 2026-09-02T06:00:00.000Z https://lambdia.com/a/same-rate-but-take-the-forward 2026-09-02T09:00:00.000Z https://lambdia.com/a/gamma-explodes-at-the-kink 2026-08-27T12:00:00.000Z https://lambdia.com/a/a-straddle-wants-a-jump-not-a-drift 2026-08-27T18:00:00.000Z https://lambdia.com/a/recombining-tree-collapses-exponential-growth 2026-08-28T06:00:00.000Z https://lambdia.com/a/garch-remembers-yesterdays-shock 2026-08-28T09:00:00.000Z https://lambdia.com/a/the-volatility-smile-breaks-the-model 2026-08-28T12:00:00.000Z https://lambdia.com/a/convexity-recovers-almost-all-the-error 2026-08-28T15:00:00.000Z https://lambdia.com/a/two-barriers-are-worth-less-than-two-options 2026-08-28T18:00:00.000Z https://lambdia.com/a/the-empty-field-has-the-higher-forward 2026-08-29T06:00:00.000Z https://lambdia.com/a/the-perpetual-put-has-a-fixed-trigger 2026-09-02T12:00:00.000Z https://lambdia.com/a/a-fat-coupon-flips-forward-premium 2026-09-02T15:00:00.000Z https://lambdia.com/a/only-a-fixed-jump-can-be-hedged 2026-09-02T18:00:00.000Z https://lambdia.com/a/matched-duration-cancels-the-level 2026-09-03T06:00:00.000Z https://lambdia.com/a/one-subtraction-prices-the-double-barrier 2026-09-03T09:00:00.000Z https://lambdia.com/a/a-call-on-the-square-kinks-at-ten 2026-09-03T12:00:00.000Z https://lambdia.com/a/vols-add-like-arrows-not-numbers 2026-09-03T15:00:00.000Z https://lambdia.com/a/a-shifted-market-line-is-a-missing-factor 2026-09-03T18:00:00.000Z https://lambdia.com/a/an-infinite-sum-that-stays-under-two 2026-09-04T06:00:00.000Z https://lambdia.com/a/it-settles-at-one-not-at-zero 2026-09-04T09:00:00.000Z https://lambdia.com/a/the-linear-term-just-moves-the-bell 2026-09-04T12:00:00.000Z https://lambdia.com/a/only-constants-stay-bounded-on-the-plane 2026-09-04T15:00:00.000Z https://lambdia.com/a/the-call-is-four-tenths-of-the-swing 2026-09-04T18:00:00.000Z https://lambdia.com/a/eight-lilies-save-only-three-days Eight Water Lilies Buy Three Days, and Dividing Says Twenty-Six https://i.ytimg.com/vi/a8Q-qpkUrvA/hqdefault.jpg One lily doubling daily covers the pond on day thirty, so eight lilies must finish in 30/8 = 3.75 days. They finish on day twenty-seven, because eight is two cubed and that slides the whole schedule exactly three days earlier. The article carries the general rule that k lilies save the floor of log base two of k, the case where five lilies save only two, and the non-overlap assumption the answer quietly rests on. https://www.youtube-nocookie.com/embed/a8Q-qpkUrvA 527 2026-08-03T14:00:00Z yes Lambdia 2026-08-03T14:00:00.000Z https://lambdia.com/a/mcdonalds-count-from-your-own-town 2026-08-25T18:00:00.000Z https://lambdia.com/a/american-call-is-worth-more-alive 2026-08-26T06:00:00.000Z https://lambdia.com/a/deterministic-and-still-unpredictable 2026-08-26T09:00:00.000Z https://lambdia.com/a/gas-stations-from-cars-per-pump 2026-08-26T12:00:00.000Z https://lambdia.com/a/us-rates-still-move-a-mexican-bond 2026-08-26T15:00:00.000Z https://lambdia.com/a/five-hundred-futures-halve-the-exposure 2026-08-26T18:00:00.000Z https://lambdia.com/a/digital-option-is-a-discounted-probability 2026-08-27T06:00:00.000Z https://lambdia.com/a/one-strange-swap-is-really-three 2026-08-27T09:00:00.000Z https://lambdia.com/a/barbers-in-chicago-from-haircuts 2026-08-29T09:00:00.000Z https://lambdia.com/a/n-of-d1-is-a-share-count-not-a-chance 2026-08-29T12:00:00.000Z https://lambdia.com/a/bond-price-curve-must-bend-upward 2026-08-29T15:00:00.000Z https://lambdia.com/a/down-and-out-calls-have-steeper-deltas 2026-08-29T18:00:00.000Z https://lambdia.com/a/hedging-that-loses-on-both-legs 2026-08-30T06:00:00.000Z https://lambdia.com/a/three-curves-of-one-call-price 2026-08-30T09:00:00.000Z https://lambdia.com/a/car-batteries-from-fleet-and-lifetime 2026-08-30T12:00:00.000Z https://lambdia.com/a/averaging-makes-asian-options-cheaper 2026-08-30T15:00:00.000Z https://lambdia.com/a/time-pushes-an-itm-delta-to-one 2026-08-30T18:00:00.000Z https://lambdia.com/a/certain-upside-calls-for-the-forward 2026-08-31T06:00:00.000Z https://lambdia.com/a/stop-loss-hedging-gets-whipsawed 2026-08-31T09:00:00.000Z https://lambdia.com/a/sizing-a-can-plant-for-ohio 2026-08-31T12:00:00.000Z https://lambdia.com/a/one-unnamed-insider-dries-up-volume 2026-08-31T15:00:00.000Z https://lambdia.com/a/one-over-root-n-is-the-poll-margin Three Points from a Thousand People, and the Two Roundings That Cancel https://i.ytimg.com/vi/QD709NCv8B0/hqdefault.jpg The 95 percent margin on a proportion is almost exactly 1 over the square root of the sample size, because p(1-p) is flat enough near its peak to call a quarter and 1.96 is close enough to 2, and those two roundings are reciprocal so they annihilate. At N = 1000 the shortcut gives 3.16 percent against an exact 3.04, and it always errs on the conservative side. Reporting one standard error instead, 1.55 percent, describes a 68 percent interval rather than a 95 percent one. https://www.youtube-nocookie.com/embed/QD709NCv8B0 50 2026-08-17T18:00:00Z yes Lambdia 2026-08-17T18:00:00.000Z https://lambdia.com/a/six-month-vol-is-sixty-over-root-two Forty-Two Dollars in Six Months, and the Quarter Million That Is Not There https://i.ytimg.com/vi/5isJ3MkzSf8/hqdefault.jpg Six months of a sixty dollar year carries 60 over root two, which is 42.43 rather than 30, because variances add over disjoint intervals and standard deviations do not, so the digital is worth exactly $239,750. The figure of $250,000 in circulation comes from rounding the z score 0.7071 up to 0.75 and then reading the tail at 0.75 as 0.25, but Phi(0.75) = 0.773373, so even the rounded chain gives 0.2266. Rounding z upward has to make the tail smaller, and 0.25 is larger, which is the tell that a symbol changed meaning mid-calculation. https://www.youtube-nocookie.com/embed/5isJ3MkzSf8 59 2026-08-18T06:00:00Z yes Lambdia 2026-08-18T06:00:00.000Z https://lambdia.com/a/lighthouse-beam-hits-pi-miles-per-second A Lighthouse Beam That Sweeps the Shore at Pi Miles a Second https://i.ytimg.com/vi/Zcp81CmKpV4/hqdefault.jpg Differentiating y = L tan(wt) gives a spot speed of w R squared over L, so the footprint accelerates with the square of its distance from the lamp: a tenth of pi directly opposite, and exactly pi miles per second nine miles along. The 9 is the along-shore leg, which makes 90 the squared hypotenuse rather than the square of nine, and that misreading is the usual failure. Nothing physical moves at that speed, and a straight coast running 2310 miles would carry a nominally faster-than-light spot carrying no information at all. https://www.youtube-nocookie.com/embed/Zcp81CmKpV4 51 2026-08-18T09:00:00Z yes Lambdia 2026-08-18T09:00:00.000Z https://lambdia.com/a/a-pizza-for-eight-is-13-86-inches A Pizza for Eight Is 13.86 Inches, Not Sixteen https://i.ytimg.com/vi/PIiDJ4ELCnA/hqdefault.jpg Servings follow area and area follows the square of the width, so feeding eight instead of six multiplies the diameter by the square root of four thirds: exactly 8 root 3, or 13.8564 inches, about 15.5 percent wider. Sixteen inches carries 16/9 of the area and would feed 10.67 people, so the reflex over-orders by nearly three servings. Allowing a one inch bare crust moves the answer down to 13.55, because a wider pizza spends proportionally less of itself on edge. https://www.youtube-nocookie.com/embed/PIiDJ4ELCnA 48 2026-08-18T12:00:00Z yes Lambdia 2026-08-18T12:00:00.000Z https://lambdia.com/a/outrun-a-dog-four-times-faster Outrunning a Dog Four Times Faster, on the Margin Between Pi and Three https://i.ytimg.com/vi/srD51s0oPn0/hqdefault.jpg Running straight out from the centre loses, because a radius costs you 1 while half the fence costs the dog pi over 4. Inside a quarter of the radius your angular speed beats his, so you can orbit until he is diametrically opposite and then sprint three quarters of a radius against his pi over 4, and the whole escape reduces to 3 being less than pi with a margin of 0.0354 R. That two-phase plan works only up to a speed ratio of pi + 1, while the best known strategy for the problem reaches 4.60334. https://www.youtube-nocookie.com/embed/srD51s0oPn0 47 2026-08-18T15:00:00Z yes Lambdia 2026-08-18T15:00:00.000Z https://lambdia.com/a/free-fall-400-metres-in-nine-seconds Four Hundred Metres in Nine Seconds, and the Answer That Is Exactly Half https://i.ytimg.com/vi/DBcSvbAS0xY/hqdefault.jpg With no air, v squared equals 2gD gives 89.4 metres per second and t equals root of 2D/g gives 8.94 seconds, both stable under g = 10 or g = 9.81. Dividing the height by the impact speed returns 4.47 seconds, wrong by exactly a factor of two at every drop height, because a body released from rest averages half its final speed. Real air reverses the picture: a coin-sized disc reaches terminal velocity near 11.9 metres per second and takes about 34.6 seconds, so nine seconds is a floor and 200 miles an hour a ceiling. https://www.youtube-nocookie.com/embed/DBcSvbAS0xY 51 2026-08-18T18:00:00Z yes Lambdia 2026-08-18T18:00:00.000Z https://lambdia.com/a/double-the-days-multiply-by-root-two Twice the Days, One Point Four One Times the Price https://i.ytimg.com/vi/X4ItBpqMfmg/hqdefault.jpg A contract struck at the current share price is worth roughly 0.3989 S sigma root T, so doubling the time to expiry multiplies the price by root two and takes 100 dollars to about 141 rather than 200. The exact ratio erf(s/2) over erf(s/(2 root 2)) is always strictly below root two because erf is concave, so 141.42 is a ceiling never reached. Strip out the strike condition and the rule collapses: the same doubling multiplies a strike 30 percent above spot by 4.19 and one 30 percent below by 1.01. https://www.youtube-nocookie.com/embed/X4ItBpqMfmg 49 2026-08-19T06:00:00Z yes Lambdia 2026-08-19T06:00:00.000Z https://lambdia.com/a/gaussian-integral-is-root-pi Root Pi from a Bell Curve, by Leaving the Number Line https://i.ytimg.com/vi/4spLl97sMoY/hqdefault.jpg There is no elementary antiderivative to evaluate, and the checkable slice of that is one line: if p is a polynomial then p' - 2xp has degree deg p + 1, which can never equal the degree of 1. Squaring the integral turns it into a rotationally symmetric integral over the plane, where the polar area element supplies the factor r that makes the radial integral elementary, so I squared equals 2 pi times one half. The same idea survives without polar coordinates via the substitution y = xt, and it fails for e to the minus x to the fourth because x^4 + y^4 is not a function of the radius. https://www.youtube-nocookie.com/embed/4spLl97sMoY 50 2026-08-19T09:00:00Z yes Lambdia 2026-08-19T09:00:00.000Z https://lambdia.com/a/derivative-of-x-to-the-x The Slope of x to the x Is Both Wrong Answers Added https://i.ytimg.com/vi/mhNjonq_GNE/hqdefault.jpg Logarithmic differentiation turns the exponent into a factor and gives x^x times (1 + ln x), which is exactly the sum of the power-rule answer x^x and the exponential-rule answer x^x ln x. That is a theorem rather than a coincidence: the two rules are the partial derivatives of u^v, and walking the diagonal u = v = x adds both partial effects. The power rule accidentally returns the correct slope at x = 1, which is the one point nobody should use to test a rule. https://www.youtube-nocookie.com/embed/mhNjonq_GNE 54 2026-08-19T12:00:00Z yes Lambdia 2026-08-19T12:00:00.000Z https://lambdia.com/a/clock-hands-meet-at-three-sixteen Sixteen and Four Elevenths Minutes Past Three https://i.ytimg.com/vi/YBji3GfEMOo/hqdefault.jpg The long hand turns 6 degrees a minute and the short one half a degree, so a 90 degree gap closes at 5.5 degrees a minute and the hands coincide 180/11 minutes past three, at 3:16:21.8181. At 3:15 the long hand has reached the 3 and the short hand is 7.5 degrees ahead of it, which is the whole content of the wrong answer. Consecutive coincidences are 720/11 minutes apart, so there are eleven per twelve hours and twenty-two per day, and the common phrasing "eleven times a day" is wrong by a factor of two. https://www.youtube-nocookie.com/embed/YBji3GfEMOo 54 2026-08-19T15:00:00Z yes Lambdia 2026-08-19T15:00:00.000Z https://lambdia.com/a/twelve-marbles-three-weighings Twelve Marbles, Three Weighings, and Twenty-Seven Ways to Land https://i.ytimg.com/vi/0qICqQZN2Ws/hqdefault.jpg Twenty-four possible answers against 3^3 = 27 outcome sequences leaves just enough room, and four against four splits those answers into exactly 8, 8 and 8. Six against six always tips, so it wastes the level outcome and leaves twelve answers for nine remaining sequences, which makes halving impossible rather than merely slow. The counting argument bounds outcome sequences rather than strategies, so it rules out every adaptive continuation at once, and the explicit schedule closes the positive half. https://www.youtube-nocookie.com/embed/0qICqQZN2Ws 48 2026-08-19T18:00:00Z yes Lambdia 2026-08-19T18:00:00.000Z https://lambdia.com/a/the-implied-forward-rate-is-20-227-percent The Back Half Pays 20.227 Percent, and the Fifth Root Cancels https://i.ytimg.com/vi/tk9F--UC-Lk/hqdefault.jpg Because the horizons are ten and five, both sides of the no-arbitrage equation are fifth powers and the root disappears, leaving 1 + f = 1.15 squared over 1.10 = 529/440, so f = 89/440 exactly. Reflecting 10 percent around 15 to get 20 is low by exactly (b - a) squared over (1 + a), a square over a positive number, which is why the reflection can never overshoot for any pair of rates. Under continuous compounding the same problem is linear and 20 percent is exactly right, so the instinct is correct machinery pointed at the wrong convention. https://www.youtube-nocookie.com/embed/tk9F--UC-Lk 51 2026-08-20T06:00:00Z yes Lambdia 2026-08-20T06:00:00.000Z https://lambdia.com/a/same-average-a-thousand-times-less-risk The Same Three and a Half Million, with a Thousandth of the Spread https://i.ytimg.com/vi/8Ely0bS-l0Y/hqdefault.jpg Both games pay 3.5 million dollars on average, so "they match" is true and the inference that it is a wash is not. Shrinking the ticket by a million divides the spread by a million while adding a million independent rolls multiplies it back by only a thousand, so the ratio of standard deviations is exactly root of a million: 1,707,825 against 1,707.83. The whole argument rests on independence, and at a correlation of 1 the diversified game becomes the single roll exactly. https://www.youtube-nocookie.com/embed/8Ely0bS-l0Y 46 2026-08-20T09:00:00Z yes Lambdia 2026-08-20T09:00:00.000Z https://lambdia.com/a/four-years-of-vol-is-only-double Four Years of Risk Is Twenty Percent, Not Forty https://i.ytimg.com/vi/B2NbQmOQzWs/hqdefault.jpg The standard deviation of a sum is not the sum of the standard deviations, so quadrupling the horizon only doubles the risk. The article carries the general square-root law, the ratio that diagnoses the mistake, and the controls showing a bell curve does none of the work: a two-point yearly return lands on 0.20026 and a uniform one on 0.20008. It also carries what actually breaks the rule, which is dependence rather than fat tails. https://www.youtube-nocookie.com/embed/B2NbQmOQzWs 55 2026-08-20T12:00:00Z yes Lambdia 2026-08-20T12:00:00.000Z https://lambdia.com/a/missing-number-from-one-sum 5050 Minus the Total, in One Pass https://i.ytimg.com/vi/c35dCKzIs1k/hqdefault.jpg Sorting the list finds the gap and is merely wasteful, which is why the article says so rather than striking it out. The subtraction works because 5050 is a closed form available before the list is read, and the two conditions carrying it are distinctness and a known range. The article adds the duplicate-hunting mirror image, the sum-of-squares route when two values are absent, and the exclusive-or accumulator for when the total would overflow. https://www.youtube-nocookie.com/embed/c35dCKzIs1k 46 2026-08-20T15:00:00Z yes Lambdia 2026-08-20T15:00:00.000Z https://lambdia.com/a/red-black-card-game-worth-2-62 The Right to Stop a Balanced Deck Is Worth $2.62 https://i.ytimg.com/vi/aP5NeY7EQgc/hqdefault.jpg Turning all fifty-two cards lands on exactly zero, which makes zero the floor rather than the value. Backward induction over the grid of remaining cards gives the exact rational 41984711742427/15997372030584, and a two-line argument shows the optimal policy can never finish below zero in any deal. The article carries the small-deck ladder, the stopping boundary the table actually produces, and two plausible rules that lose money against it. https://www.youtube-nocookie.com/embed/aP5NeY7EQgc 52 2026-08-20T18:00:00Z yes Lambdia 2026-08-20T18:00:00.000Z https://lambdia.com/a/atm-delta-is-above-one-half An At-the-Money Call's Delta Is Never Exactly a Half https://i.ytimg.com/vi/QVUUKNHXzNQ/hqdefault.jpg At the money the log term in d1 vanishes and what remains is strictly positive for every non-negative rate and every volatility, so the delta always beats 0.5 and is 0.6554 at twenty percent. A square rather than a derivative gives the sharp floor: at six percent over a year the delta can never fall below 0.6355. The article also kills the sentence that sounds like a restatement of the answer, since the chance of finishing in the money falls to 0.4801 at forty percent volatility. https://www.youtube-nocookie.com/embed/QVUUKNHXzNQ 43 2026-08-21T06:00:00Z yes Lambdia 2026-08-21T06:00:00.000Z https://lambdia.com/a/all-four-right-is-one-in-sixteen Ten to One on Four Coin Calls Costs 31 Cents on the Dollar https://i.ytimg.com/vi/VOfmfuHGAc8/hqdefault.jpg One chance in sixteen needs fifteen to one to break even, so a ten-to-one ticket is priced as though the calls came right nine times in a hundred rather than six and a quarter. The fair payout doubles and adds one with every leg, which is why multi-leg tickets run away from any quote a seller offers. The article carries the noise that hides the loss, 2.663 of spread against 0.3125 of edge, and the five-point edge per leg that would flip the verdict. https://www.youtube-nocookie.com/embed/VOfmfuHGAc8 45 2026-08-21T09:00:00Z yes Lambdia 2026-08-21T09:00:00.000Z https://lambdia.com/a/correlation-ignores-shifts-and-scalings Slide It, Stretch It, and the Correlation Does Not Move https://i.ytimg.com/vi/4MT3VKtORBU/hqdefault.jpg A shift leaves every deviation from the mean untouched, and a stretch multiplies the covariance and one standard deviation by the same factor, so both cancel out of the ratio. The tempting answer of five times rho is worse than wrong: at rho = 0.40 it names 2.0, which Cauchy-Schwarz forbids any correlation from reaching. The article carries the general affine rule, the sign flip a negative factor produces, and the curved maps the invariance does not survive. https://www.youtube-nocookie.com/embed/4MT3VKtORBU 45 2026-08-21T12:00:00Z yes Lambdia 2026-08-21T12:00:00.000Z https://lambdia.com/a/the-trade-that-loses-four-times-in-five A Contract That Loses Four Times in Five and Is Worth 1.80 https://i.ytimg.com/vi/VJLZno2-eRE/hqdefault.jpg Four settlements in five come back below the 1.50 outlay, and the average payoff is still 1.80, an edge of 0.30 a contract or twenty percent of the money at risk. The reflex is not bad arithmetic, it is the mode standing in for the mean. The article carries the tally over one full cycle, the threshold saying you need the large outcome more often than one time in eight, and the reason waiting longer can leave you less likely to be ahead. https://www.youtube-nocookie.com/embed/VJLZno2-eRE 45 2026-08-21T15:00:00Z yes Lambdia 2026-08-21T15:00:00.000Z https://lambdia.com/a/85-7-percent-in-the-calmer-stock The Calmest Blend of a 20% and a 30% Stock Swings 19.64% https://i.ytimg.com/vi/4OdOuXhzW94/hqdefault.jpg Six sevenths in the quieter share beats holding it alone, because what the jumpy share adds at the margin is its correlation times its own swing, fifteen against twenty. The derivative of the variance at a full allocation is exactly +1/50, so the informal test and the first-order condition are one statement. The article carries the exact optimum sqrt(27/700), the convexity making it a minimum, and the correlation threshold of two thirds above which the dip disappears entirely. https://www.youtube-nocookie.com/embed/4OdOuXhzW94 47 2026-08-21T18:00:00Z yes Lambdia 2026-08-21T18:00:00.000Z https://lambdia.com/a/take-the-three-pointer A 40% Shot Beats a 70% Shot That Only Buys a Coin Flip https://i.ytimg.com/vi/ivcda1CbrMc/hqdefault.jpg Going in and winning are different events: the short shot clears two hurdles and wins 0.35 of the time against the long shot's 0.40. The article prices how wrong the reflex is in two currencies, a break-even overtime rate of 4/7 and a break-even make rate of 80 percent at a coin-flip overtime. It also names the objective under which the reflex is right, since the short shot scores 1.40 expected points against 1.20 and still wins fewer games. https://www.youtube-nocookie.com/embed/ivcda1CbrMc 49 2026-08-22T06:00:00Z yes Lambdia 2026-08-22T06:00:00.000Z https://lambdia.com/a/the-call-beats-the-put-ten-percent-out Ten Percent Up Is a Smaller Move Than Ten Percent Down https://i.ytimg.com/vi/qT2I93pzMlM/hqdefault.jpg The put reaches its strike more often, 0.3348 against 0.2821, and the call is still worth more, 4.2920 against 3.5891. The mechanism is not the unbounded-upside story, which would predict a gap at the money where put-call parity provably gives none; at a zero rate the 110 call equals 1.1 times a put struck at 90.909, and the put on offer is struck lower than that. The article also records two circulating claims that fail at these strikes, since the in-the-money chances at r = sigma^2/2 are 0.3168 and 0.2992 rather than equal, and the price ratio is 1.63 rather than 2. https://www.youtube-nocookie.com/embed/qT2I93pzMlM 47 2026-08-22T09:00:00Z yes Lambdia 2026-08-22T09:00:00.000Z https://lambdia.com/a/square-root-of-204000-by-hand 451.67 From One Division, and Always a Shade Too High https://i.ytimg.com/vi/bhjHKk-AoL4/hqdefault.jpg The anchor is 45 squared, the shortfall is 1500, and one division by 900 lands on 1355/3 = 451.6667 against a true 451.66359. The estimate overshoots by exactly h squared over four a squared, which is 25/9 in the square here, so the error has a known sign as well as a known size. The article carries the bracket that names 452 as the nearest integer, one Newton step to nine figures, and what happens when the anchor is chosen too far away. https://www.youtube-nocookie.com/embed/bhjHKk-AoL4 48 2026-08-22T12:00:00Z yes Lambdia 2026-08-22T12:00:00.000Z https://lambdia.com/a/two-envelopes-swap-gains-nothing The 25% Gain From Swapping Needs a Distribution That Does Not Exist https://i.ytimg.com/vi/QDzRngiZh3o/hqdefault.jpg Requiring the fifty-fifty at every amount you could open forces the weights to satisfy f(x) = f(x/2)/2, whose only solutions are proportional to 1/x, and that integrates to infinity at both ends. Conditional on the pair, the swap gains the smaller amount or loses it with equal chance, which is zero and needs no assumption at all. The article carries a proper spread where the conditional answer is genuinely x/2, and the infinite-mean spread where swapping really is right at every observable amount. https://www.youtube-nocookie.com/embed/QDzRngiZh3o 48 2026-08-22T15:00:00Z yes Lambdia 2026-08-22T15:00:00.000Z https://lambdia.com/a/winning-the-bid-is-the-bad-news Zero Profit at Every Bid, Because Winning Says the Firm Was Cheap https://i.ytimg.com/vi/PrVqCUkKZBs/hqdefault.jpg Acceptance restricts the value to below your bid, where a uniform variable averages half of it, and doubling half your bid returns exactly your bid. The expected profit is therefore identically zero at every bid up to 100 and 100 minus b above it, so there is no optimal bid to find. With a general multiplier the profit is b squared times k minus 2, over 200, making doubling the exact break-even multiple, and the article shows a value distribution starting at 50 where the same bidder profits. https://www.youtube-nocookie.com/embed/PrVqCUkKZBs 48 2026-08-22T18:00:00Z yes Lambdia 2026-08-22T18:00:00.000Z https://lambdia.com/a/black-scholes-in-ten-seconds Pricing an Option in Your Head, and the 0.4 Nobody Explains https://i.ytimg.com/vi/EBtWqYdHwzU/hqdefault.jpg A three-month at-the-money call on a stock at 100 with 40% volatility is worth about eight dollars, and you can get there in two multiplications. The constant four tenths turns out to be the height of the normal bell at its peak, and the whole error of the mental rule is one rounding plus one cubic term. Scaling volatility linearly with time instead of with its square root gives ten dollars, which is 25.5% too high. https://www.youtube-nocookie.com/embed/EBtWqYdHwzU 54 2026-08-17T06:00:00Z yes Lambdia 2026-08-17T06:00:00.000Z https://lambdia.com/a/fair-thirds-from-a-fair-coin Three Children, One Coin, and Eight Thirds of a Flip https://i.ytimg.com/vi/329RzlHQHbw/hqdefault.jpg A fair coin cuts probabilities into halves and quarters, and a short argument about the prime factorisation of two shows it can never reach one third in a bounded number of flips. Dropping the bound fixes it: flip twice, bin the tail-tail, and each child holds exactly a third for 8/3 flips on average. That naive scheme turns out to be the best any coin-flipping procedure can do for three outcomes, which stops being true at five. https://www.youtube-nocookie.com/embed/329RzlHQHbw 52 2026-08-17T09:00:00Z yes Lambdia 2026-08-17T09:00:00.000Z https://lambdia.com/a/standard-deviation-of-one-to-five Five Numbers, Two Answers, and the Divisor Nobody Asks About https://i.ytimg.com/vi/g4qKbTuo6_U/hqdefault.jpg The standard deviation of 1, 2, 3, 4, 5 is either 1.4142 or 1.5811, and offering one of them without asking which question you are answering is the only wrong move. The sum of squared deviations is 10 either way, so everything turns on whether you divide it by 5 or by 4. Bessel's correction makes the variance unbiased and leaves the standard deviation biased low by about six percent at this sample size, and a third divisor beats both of them if you optimise for mean squared error instead. https://www.youtube-nocookie.com/embed/g4qKbTuo6_U 60 2026-08-17T12:00:00Z yes Lambdia 2026-08-17T12:00:00.000Z https://lambdia.com/a/one-dollar-at-a-barrier-costs-75-cents A Dollar at the Barrier, Seventy-Five Cents Today https://i.ytimg.com/vi/sL1JOMj4dVI/hqdefault.jpg A share sits at 75, the rate is zero, and a perpetual claim pays one dollar the first time the price ever touches 100. It is worth exactly 75 cents, and no volatility number is needed to say so. The reflex answer of a dollar assumes the barrier is always reached, which a price with a floor at zero never promises: a quarter of the paths fade away without paying anything. https://www.youtube-nocookie.com/embed/sL1JOMj4dVI 51 2026-08-17T15:00:00Z yes Lambdia 2026-08-17T15:00:00.000Z https://lambdia.com/a/stop-on-five-or-six-and-win-4-67 Three Rolls, One Decision, and 14/3 https://i.ytimg.com/vi/9Y5Mq837ZX8/hqdefault.jpg A die is rolled up to three times and you are paid the face you stop on. The reflex answer of 3.5 is the value of the same game with the right to stop deleted, and the real value is 14/3, reached by computing the game from its last roll backwards. The thresholds move as rolls run out, which is why a four is worth keeping late and worth rejecting early. https://www.youtube-nocookie.com/embed/9Y5Mq837ZX8 47 2026-08-15T18:00:00Z yes Lambdia 2026-08-15T18:00:00.000Z https://lambdia.com/a/atm-call-and-put-cost-the-same Unlimited Upside, Identical Price https://i.ytimg.com/vi/wpImRJ6jKJ4/hqdefault.jpg An at-the-money call has no ceiling on its payoff and an at-the-money put is capped at the strike, yet at a zero interest rate the two cost exactly the same. The reason is put-call parity and it uses no model at all: the difference of the two payoffs is a straight line, so pricing it needs only the risk-neutral mean. The equality was checked on five terminal distributions with mean at the strike, and on a sixth whose mean is 120, where the gap is exactly 20. https://www.youtube-nocookie.com/embed/wpImRJ6jKJ4 52 2026-08-16T06:00:00Z yes Lambdia 2026-08-16T06:00:00.000Z https://lambdia.com/a/third-dial-number-is-free Forty Cubed, Forty Squared, and the Number You Never Choose https://i.ytimg.com/vi/o-97MEod4Bs/hqdefault.jpg A safe takes three numbers from a dial marked 1 to 40, so there are 64,000 combinations, and the worst case is 1600 attempts rather than 64,000. The third number is supplied by the mechanism instead of guessed, which collapses the search from three dimensions to two, and 1600 is proved both achievable and unavoidable. A dial with a mark of mechanical slack drops the count to 196, which is a covering problem on a cycle of forty. https://www.youtube-nocookie.com/embed/o-97MEod4Bs 51 2026-08-16T09:00:00Z yes Lambdia 2026-08-16T09:00:00.000Z https://lambdia.com/a/ages-two-two-and-nine Three Ages, a Sum She Cannot Use, and 2, 2 and 9 https://i.ytimg.com/vi/CnOsjC406v4/hqdefault.jpg Three children's ages multiply to 36. Someone who knows the sum admits she cannot name them, and that admission is the only real clue in the problem. Eight triples, one repeated sum, and a second clue that eliminates nothing on its own yet decides everything once the first has run. https://www.youtube-nocookie.com/embed/CnOsjC406v4 56 2026-08-15T09:00:00Z yes Lambdia 2026-08-15T09:00:00.000Z https://lambdia.com/a/the-mirror-flips-depth-not-left-right A Mirror Reverses One Axis, and It Is Not Left or Right https://i.ytimg.com/vi/N70aAOFuI5k/hqdefault.jpg Raise your right hand at a mirror and the reflected hand stays on the same side of the room, which means the usual question has a false premise. A plane mirror is the matrix diag(1, 1, -1): it fixes both axes lying in the glass and reverses only the direction you look along. Its determinant is -1, so no rotation reproduces it, and the sideways flip everyone reports belongs to the half turn you perform in your head. https://www.youtube-nocookie.com/embed/N70aAOFuI5k 49 2026-08-15T12:00:00Z yes Lambdia 2026-08-15T12:00:00.000Z https://lambdia.com/a/inscribed-circle-forces-side-fifty A Circle in a Square, a 5 by 10 Rectangle, and the Root You Throw Away https://i.ytimg.com/vi/0uqW7kZoA-8/hqdefault.jpg One radius drawn to the rectangle's far corner turns the whole problem into a right triangle with legs R - 10 and R - 5, and the quadratic that follows has roots 5 and 25. Both satisfy the equation exactly, so rejecting 5 takes geometry rather than arithmetic: at that radius the corner really does touch the circle while the rectangle has already swallowed half the disk. The general a by b rectangle shows the discarded root is a permanent feature of squaring. https://www.youtube-nocookie.com/embed/0uqW7kZoA-8 54 2026-08-15T15:00:00Z yes Lambdia 2026-08-15T15:00:00.000Z https://lambdia.com/a/nine-digits-four-sums-of-twenty Nine Digits, Four Windows, and a Middle That Cannot Move https://i.ytimg.com/vi/Pq_e65R1aME/hqdefault.jpg Nine slots hold the digits 1 to 9 and four overlapping windows each add to 20. One subtraction pins the centre digit to 5 with no search at all, and the same structure classifies every arrangement: there are 96 of them, falling into 48 mirror pairs. Change the target and the centre is forced to an odd digit that sometimes admits nothing. https://www.youtube-nocookie.com/embed/Pq_e65R1aME 51 2026-08-16T12:00:00Z yes Lambdia 2026-08-16T12:00:00.000Z https://lambdia.com/a/tenth-morning-means-ten The Announcement That Told Nobody Anything https://i.ytimg.com/vi/JRcVz_AxDCA/hqdefault.jpg A stranger says out loud what every islander can already see, and ten days later ten people leave. The fact was mutual knowledge all along; what the announcement supplied was the nine levels of nested knowledge above it. An explicit count over 4096 possible worlds settles the induction without trusting it. https://www.youtube-nocookie.com/embed/JRcVz_AxDCA 53 2026-08-16T15:00:00Z yes Lambdia 2026-08-16T15:00:00.000Z https://lambdia.com/a/first-coin-in-the-centre-wins The Coin in the Middle, and the Table Where It Fails https://i.ytimg.com/vi/q2-lDBBvX-k/hqdefault.jpg Take the centre, then mirror every move through it, and you place the last coin. The proof has three requirements and only one of them needs that opening move, which is the step a one-line answer skips. Central symmetry alone is not the condition: an annulus is centrally symmetric and the first player loses on it. https://www.youtube-nocookie.com/embed/q2-lDBBvX-k 49 2026-08-16T18:00:00Z yes Lambdia 2026-08-16T18:00:00.000Z https://lambdia.com/a/weigh-a-plane-with-tyre-pressure Weighing a Parked Jet With a Tyre Gauge, and Why the Answer Is a Bracket https://i.ytimg.com/vi/3kXbowspXK0/hqdefault.jpg The reflex answer is that nothing can be weighed without a scale, and it is wrong: an aircraft resting on inflated tyres is already standing on four scales, each with a dial on it. Pressure times contact patch gives 160,000 pounds, but the deliverable is the interval from 115,200 to 211,200 together with the direction of the bias. A stiff sidewall carries part of the load, so the reading is a floor rather than a measurement. https://www.youtube-nocookie.com/embed/3kXbowspXK0 52 2026-08-23T06:00:00Z yes Lambdia 2026-08-23T06:00:00.000Z https://lambdia.com/a/when-diversification-stops-paying The Correlation Equals the Ratio of the Swings, and Diversification Stops Paying https://i.ytimg.com/vi/TaaYzXYXNuY/hqdefault.jpg Two stocks with equal expected returns, variances 0.10 and 0.40, and correlation 0.5: the reflex differentiates the portfolio variance and reports an interior weight. The minimum sits at 100% in the calmer stock, and the usual explanation for that, which blames the no-shorting rule, is wrong. The vertex of the variance parabola lands exactly on w = 1, so the constraint does no work at all and the answer survives dropping it. https://www.youtube-nocookie.com/embed/TaaYzXYXNuY 49 2026-08-23T09:00:00Z yes Lambdia 2026-08-23T09:00:00.000Z https://lambdia.com/a/ping-pong-balls-inside-a-747 How Many Ping-Pong Balls Fill a Jumbo, and Why Thirty Million Is the One Impossible Answer https://i.ytimg.com/vi/71dyLAFzF28/hqdefault.jpg Dividing the cabin by the ball gives 29.8 million, which is exact arithmetic on the assumption that spheres tile space. They do not, and the correction is pinned on both sides by constants: a plain cubic grid anyone can build holds exactly 15,625,000 balls, and no arrangement whatever beats pi over root eighteen, which caps the count at 22.1 million. The answer is that interval, with a settled pour at 19.1 million sitting inside it. https://www.youtube-nocookie.com/embed/71dyLAFzF28 49 2026-08-23T12:00:00Z yes Lambdia 2026-08-23T12:00:00.000Z https://lambdia.com/a/game-one-decides-five-times-in-sixteen Game One Is Worth 31.25 of Your 100, and Nobody Asked Who Is Better https://i.ytimg.com/vi/hzczcSSZ8gw/hqdefault.jpg A holding pays $200 if a team wins four games first, you must take a symmetric position on every game, and committing the whole hundred to game one produces the right payoffs a week too early. Backward induction on the lattice fixes the amount at half the gap between the two successor values, $31.25. The same number is 5/16 of the holding, which is the chance the other six games split three each, and no win probability appears anywhere in the derivation. https://www.youtube-nocookie.com/embed/hzczcSSZ8gw 45 2026-08-23T15:00:00Z yes Lambdia 2026-08-23T15:00:00.000Z https://lambdia.com/a/bachelier-priced-options-in-1900 The 1900 Option Formula: Why an At-the-Money Call Is Worth Two Fifths of a Swing https://i.ytimg.com/vi/Bc7krQTLG6s/hqdefault.jpg A share swinging twenty dollars a year gives an at-the-money call that looks like it should cost ten, half the swing collected half the time. It costs 7.98, because the upper half of a bell curve averages 0.798 of a standard deviation rather than a whole one. The article derives the general arithmetic-Brownian price, checks both limits, and quantifies the negative-price defect that got the model retired and then rehabilitated. https://www.youtube-nocookie.com/embed/Bc7krQTLG6s 51 2026-08-23T18:00:00Z yes Lambdia 2026-08-23T18:00:00.000Z https://lambdia.com/a/more-vol-can-hurt-a-digital 2026-08-24T06:00:00.000Z https://lambdia.com/a/fish-in-the-ocean-by-volume 2026-08-24T09:00:00.000Z https://lambdia.com/a/ito-subtracts-t-over-two 2026-08-24T12:00:00.000Z https://lambdia.com/a/same-expectation-lower-variance 2026-08-24T15:00:00.000Z https://lambdia.com/a/theta-and-gamma-point-opposite-ways 2026-08-24T18:00:00.000Z https://lambdia.com/a/one-head-makes-the-bias-density-2p 2026-08-25T06:00:00.000Z https://lambdia.com/a/time-integral-of-brownian-is-t-cubed-over-three 2026-08-25T09:00:00.000Z https://lambdia.com/a/expected-exponential-adds-half-the-variance 2026-08-25T12:00:00.000Z https://lambdia.com/a/a-coin-flip-worth-one-dollar-eighty 2026-08-25T15:00:00.000Z https://lambdia.com/a/four-gallons-from-five-and-three Four Gallons From a Five and a Three, and Why Coprimality Is the Trick https://i.ytimg.com/vi/0-xvc2pnods/hqdefault.jpg Adding fives and threes never reaches four, and that observation is correct. It is also about the wrong set, because a pour is a subtraction and the reachable amounts are the integer combinations rather than the natural ones. An exhaustive state-graph search proves six pours is minimal, and the four missing sums turn out to be the gaps of a numerical semigroup. https://www.youtube-nocookie.com/embed/0-xvc2pnods 47 2026-08-10T06:00:00Z yes Lambdia 2026-08-10T06:00:00.000Z https://lambdia.com/a/one-weighing-finds-the-light-bag Ten Bags, One Reading: 55 Coins Turn the Dial Into a Label https://i.ytimg.com/vi/Aumr9_f8oKA/hqdefault.jpg Taking one coin from each bag reads 9.9 ounces whichever bag is light, and the failure is blindness rather than imprecision. Loading i coins from bag i makes the dial an injective function of the culprit, at a cost of the tenth triangular number. A 45-coin variant is cheaper, and powers of two identify any subset of light bags from one reading. https://www.youtube-nocookie.com/embed/Aumr9_f8oKA 53 2026-08-10T09:00:00Z yes Lambdia 2026-08-10T09:00:00.000Z https://lambdia.com/a/light-both-ends-for-thirty-seconds Thirty Seconds From a Cord That Burns Unevenly, on Any Cord At All https://i.ytimg.com/vi/nxC4-vmqnng/hqdefault.jpg Halving the length and timing the flame is not a biased estimator of half the time, it is unrelated to it: across four thousand random cords the midpoint method scattered from under fifteen seconds to over forty-five. Lighting both ends gives exactly thirty on every cord, by an argument that never evaluates the burn rate. https://www.youtube-nocookie.com/embed/nxC4-vmqnng 48 2026-08-10T12:00:00Z yes Lambdia 2026-08-10T12:00:00.000Z https://lambdia.com/a/gold-bar-in-one-two-four One Gold Bar, Two Cuts, Seven Evenings: 1, 2 and 4 https://i.ytimg.com/vi/iOYnGubdKpU/hqdefault.jpg Seven pieces cost six cuts and the schedule pays correctly, so the six-cut answer breaks one constraint and nothing else. Because the worker can hand pieces back, the contract is on his holding rather than on the transfer, and the ledger turns out to be a three-bit counter. Brute force finds 1-2-4 is the only three-piece solution. https://www.youtube-nocookie.com/embed/iOYnGubdKpU 51 2026-08-10T15:00:00Z yes Lambdia 2026-08-10T15:00:00.000Z https://lambdia.com/a/hanoi-64-rings-584-billion-years 64 Rings, and 584 Billion Years at One Move a Second https://i.ytimg.com/vi/6hL8NIW-xms/hqdefault.jpg Thirty-two rings really do take 136 years at a move a second, which is what makes doubling it to 272 so tempting. The exact ratio between the two cases factors as two to the thirty-two plus one, so the guess is short by more than four billion times. The article carries the lower bound the recursion alone does not give, and the 2-adic rule for which ring moves when. https://www.youtube-nocookie.com/embed/6hL8NIW-xms 45 2026-08-10T18:00:00Z yes Lambdia 2026-08-10T18:00:00.000Z https://lambdia.com/a/stock-drops-so-you-buy-it-back The Share Falls Ten Dollars and the Hedge Says Buy https://i.ytimg.com/vi/LZ3JRi9kXRI/hqdefault.jpg The hedge on a long call is the slope of its value, and a convex curve flattens as you slide left, so a falling share forces a smaller short and a smaller short is a purchase. No volatility, maturity or distribution enters that argument. The rebalance buys twenty shares, and the position gains 0.9824 per share on the fall. https://www.youtube-nocookie.com/embed/LZ3JRi9kXRI 49 2026-08-11T06:00:00Z yes Lambdia 2026-08-11T06:00:00.000Z https://lambdia.com/a/two-bells-ring-together-after-60-seconds Two Bells, 12 and 15 Seconds Apart, Meet at a Full Minute https://i.ytimg.com/vi/o5-dz5w1YY0/hqdefault.jpg Multiplying four rings a minute by five gives a quantity in inverse minutes squared, so a dimension check kills the twenty-second answer before any number theory. Converting to gaps makes the ring times two arithmetic progressions whose intersection is the least common multiple. Shift one bell by a second and the two never coincide at all. https://www.youtube-nocookie.com/embed/o5-dz5w1YY0 48 2026-08-11T09:00:00Z yes Lambdia 2026-08-11T09:00:00.000Z https://lambdia.com/a/bug-unfolds-the-room-root-5 A Bug Walks Root Five Across a Cubic Room https://i.ytimg.com/vi/A-fVSHIKH8o/hqdefault.jpg Unfolding two faces into a 2 by 1 rectangle turns the walk into a straight segment of length root five, crossing the shared edge at half height. The reflex answer of one plus root two is the same one-parameter family evaluated at the end of that edge instead of its middle, so the trap and the answer are two points on one curve. https://www.youtube-nocookie.com/embed/A-fVSHIKH8o 51 2026-08-11T12:00:00Z yes Lambdia 2026-08-11T12:00:00.000Z https://lambdia.com/a/water-and-wine-end-up-equal Two Jars End Up Equally Contaminated, and Conservation Proves It https://i.ytimg.com/vi/y1lfVPg0wiA/hqdefault.jpg The pour back really was diluted, and the conclusion still does not follow: both jars finish at six cups, so whatever left one jar was replaced cup for cup by what arrived. That argument needs no fractions and survives terrible stirring, while the number 1.5 cups does not. https://www.youtube-nocookie.com/embed/y1lfVPg0wiA 49 2026-08-11T15:00:00Z yes Lambdia 2026-08-11T15:00:00.000Z https://lambdia.com/a/a-hundred-heads-means-a-loaded-coin A Hundred Heads in a Row: the Answer Is Not One Half https://i.ytimg.com/vi/LsNFajEjfWU/hqdefault.jpg Coins have no memory, which is true, and nobody said this coin is fair, which is the whole problem. A fair coin explains the run with probability two to the minus one hundred while a two-headed coin explains it every time. The article locates the threshold exactly and reconciles the answer with the companion piece on ten heads, which asks a different question about a different setup. https://www.youtube-nocookie.com/embed/LsNFajEjfWU 46 2026-08-11T18:00:00Z yes Lambdia 2026-08-11T18:00:00.000Z https://lambdia.com/a/reroll-the-ones-and-expect-four Reroll the Ones and the Die Is Worth Four https://i.ytimg.com/vi/WBlntLmBR04/hqdefault.jpg Three and a half is the exact average of a plain die, which is why it survives being double-checked. The rule does not reweight six outcomes, it deletes one, leaving a uniform payoff on five faces and an answer of four. The procedure costs 1.2 rolls on average, and the version where the reroll is your choice is a different game worth 4.25. https://www.youtube-nocookie.com/embed/WBlntLmBR04 50 2026-08-12T06:00:00Z yes Lambdia 2026-08-12T06:00:00.000Z https://lambdia.com/a/two-kings-in-a-row-is-one-in-221 Two Kings Off the Top: One in 221 https://i.ytimg.com/vi/CGVPyk7QyDA/hqdefault.jpg Four over fifty-two squared is exactly right for the question where the first card goes back, which is what makes it hard to catch. Removing a king shrinks the numerator proportionally more than the denominator, and the counting route through 1326 two-card hands confirms one in 221 without mentioning order at all. https://www.youtube-nocookie.com/embed/CGVPyk7QyDA 49 2026-08-12T09:00:00Z yes Lambdia 2026-08-12T09:00:00.000Z https://lambdia.com/a/clock-face-split-into-three-26s A Clock Face in Three Pieces of 26, and No Pie Cut Ever Works https://i.ytimg.com/vi/-xOjV0ls5lA/hqdefault.jpg The dial totals 78, so each piece needs 26, and the pie instinct fails on all 220 possible cuts. The proof is three lines of triangular numbers: only one pair of running totals differs by 26, which forces the first two cracks and then demands a total of 62 that does not exist. https://www.youtube-nocookie.com/embed/-xOjV0ls5lA 55 2026-08-12T12:00:00Z yes Lambdia 2026-08-12T12:00:00.000Z https://lambdia.com/a/252-staircase-paths-to-five-five 252 Staircase Paths, and the Word That Replaces the Grid https://i.ytimg.com/vi/43yAif_FHqU/hqdefault.jpg Every route across a five by five grid is ten steps long with exactly five going east, so counting routes is choosing which five of the ten slots are east. The reflex 1024 is the exact number of free ten-step walks, and only 252 of them arrive. Forbid the route to rise above the diagonal and the count collapses to the Catalan number 42. https://www.youtube-nocookie.com/embed/43yAif_FHqU 56 2026-08-12T15:00:00Z yes Lambdia 2026-08-12T15:00:00.000Z https://lambdia.com/a/2519-leaves-every-remainder 2519, and the Shift That Turns Nine Congruences Into One https://i.ytimg.com/vi/Imed_2jogGc/hqdefault.jpg Every one of the nine conditions leaves a remainder one short of its divisor, so x plus one is divisible by all of 2 through 10 and the answer is 2519. Minimality comes free, and the whole solution set is 2520k minus 1. The article carries the coprimality caveat, the near miss 209 that satisfies six of the nine, and a variant where no shift exists. https://www.youtube-nocookie.com/embed/Imed_2jogGc 51 2026-08-12T18:00:00Z yes Lambdia 2026-08-12T18:00:00.000Z https://lambdia.com/a/pebble-needs-3-6-flips-not-4 3.6 Flips, Not 4, and the Beautiful Argument That Says Otherwise https://i.ytimg.com/vi/IBnMVPKa6eQ/hqdefault.jpg A pebble climbing four boxes on coin flips needs 18/5 flips on average, and the two-line renewal argument that gives 4 is wrong. Its premise is true, since half of all games really do end on flip two, but the non-finishing half is two different states: tails-tails sends the pebble home while heads-heads leaves it on box 3, one flip from the exit and worth only 14/5. https://www.youtube-nocookie.com/embed/IBnMVPKa6eQ 58 2026-08-13T06:00:00Z yes Lambdia 2026-08-13T06:00:00.000Z https://lambdia.com/a/two-thirds-of-a-dollar-before-the-six Two Thirds of a Dollar, and the Branch That Pays Nothing https://i.ytimg.com/vi/6eWUW3KunmA/hqdefault.jpg The pile really does average exactly one dollar, which is why almost everyone answers one dollar and why the trap is a correct calculation of the wrong quantity. The play is worth two thirds, because the roll that ends the game pays on two of its three faces and that roll is independent of how big the pile grew. https://www.youtube-nocookie.com/embed/6eWUW3KunmA 51 2026-08-13T09:00:00Z yes Lambdia 2026-08-13T09:00:00.000Z https://lambdia.com/a/say-six-and-win-the-race-to-fifty Say Six: The Only Opening That Wins the Race to Fifty https://i.ytimg.com/vi/6pogQzUIRk8/hqdefault.jpg Counting to fifty in steps of one to ten, the first player wins, and exactly one of the ten legal openings does it. The stations are 6, 17, 28, 39 and 50, spaced eleven apart because eleven is one more than the largest legal step. The article carries the residue argument that proves the opening is unique, and the target 55 where the advantage flips. https://www.youtube-nocookie.com/embed/6pogQzUIRk8 54 2026-08-13T12:00:00Z yes Lambdia 2026-08-13T12:00:00.000Z https://lambdia.com/a/three-switches-one-visit-use-heat Three Switches, One Visit, and the State Light Cannot Carry https://i.ytimg.com/vi/o9Ew4p43rh0/hqdefault.jpg Six pairings have to be separated by a single observation, and a light-only strategy always leaves at least two candidates standing. Warmth is a genuine third readable state, and three states handed to three bulbs give exactly six readings, one per pairing. The article carries the impossibility count and the four-switch case where the trick fails. https://www.youtube-nocookie.com/embed/o9Ew4p43rh0 55 2026-08-13T15:00:00Z yes Lambdia 2026-08-13T15:00:00.000Z https://lambdia.com/a/blind-man-deduces-his-hat-is-blue Blue, and the Two Sentences That Contain No Answer https://i.ytimg.com/vi/uneqvdC4oEA/hqdefault.jpg Of the eight colour triples, seven are feasible from a pool of three blue hats and two red. The first silence removes one, the second removes two more, and all four survivors put a blue hat on the third man, which is what makes his answer a deduction rather than a lucky call. A pool sweep shows three blue and two red is the only small pool where the story can happen. https://www.youtube-nocookie.com/embed/uneqvdC4oEA 55 2026-08-13T18:00:00Z yes Lambdia 2026-08-13T18:00:00.000Z https://lambdia.com/a/stopping-at-the-first-boy-keeps-fifty-fifty One Boy Per Family, One Girl Per Family, and Why ln 2 Is Not a Half https://i.ytimg.com/vi/m7lORjleAnQ/hqdefault.jpg Every family averages exactly one girl and contains exactly one boy, so the ratio of expected counts is exactly one half and a large town splits evenly. The expected share inside a single family is not one half but ln 2, and it is still 0.5249 across ten families, with the excess falling off like one over four m. https://www.youtube-nocookie.com/embed/m7lORjleAnQ 47 2026-08-14T06:00:00Z yes Lambdia 2026-08-14T06:00:00.000Z https://lambdia.com/a/pizza-choice-by-the-right-angle One Eighteen Inch Pizza Beats a Twelve and a Ten, by a Corner https://i.ytimg.com/vi/gQ9Fk5EBJdc/hqdefault.jpg The factor pi over four cancels off both sides, so the comparison is 324 against 244, about a third more pizza. Written that way it is the law of cosines: lay the three widths out as a triangle and the corner between the two smaller sides opens to 109.47 degrees, wider than square, which is the answer with no arithmetic at all. https://www.youtube-nocookie.com/embed/gQ9Fk5EBJdc 50 2026-08-14T09:00:00Z yes Lambdia 2026-08-14T09:00:00.000Z https://lambdia.com/a/fire-must-come-last-one-third One in Three, and the Card That Was Never in the Problem https://i.ytimg.com/vi/KHYHR4v3dko/hqdefault.jpg Wind appears nowhere in the winning condition, so delete it: three cards in six equally likely orders, and you win in the two where fire comes last. One in four is the correct probability that fire is last of all four cards, a strictly smaller event, and the gap is exactly one twelfth. https://www.youtube-nocookie.com/embed/KHYHR4v3dko 50 2026-08-14T12:00:00Z yes Lambdia 2026-08-14T12:00:00.000Z https://lambdia.com/a/root-x-squared-plus-x-minus-x-is-half A Half, from One Multiplication and No Calculus https://i.ytimg.com/vi/F7SRIHfNco4/hqdefault.jpg Multiplying by the conjugate turns the difference into x over the sum of the same two terms, exactly, with no series and no error term, and dividing through by x reads off the limit one half. Completing the square gives a constant excess of a quarter, so the gap climbs toward a half strictly from below and never reaches it. https://www.youtube-nocookie.com/embed/F7SRIHfNco4 53 2026-08-14T15:00:00Z yes Lambdia 2026-08-14T15:00:00.000Z https://lambdia.com/a/midpoint-of-consecutive-primes-never-prime The Middle of Two Neighbouring Primes Is Never Prime https://i.ytimg.com/vi/LjSxPk8XqmE/hqdefault.jpg The midpoint of p and q sits strictly between them, and consecutive means precisely that no prime lives in that interval, so the answer is never and the proof is two lines with no arithmetic in it. The pair 2 and 3 survives for a different reason, since five halves is not an integer, and it is the only such pair. https://www.youtube-nocookie.com/embed/LjSxPk8XqmE 49 2026-08-14T18:00:00Z yes Lambdia 2026-08-14T18:00:00.000Z https://lambdia.com/a/prime-squared-minus-one-divides-by-24 24 Divides p Squared Minus One, and Primality Is Not What Does It https://i.ytimg.com/vi/Jvg7PuWbAxA/hqdefault.jpg Factoring into p minus one times p plus one stops the question being about p: the two neighbours are consecutive even numbers so their product carries eight, and one of the three consecutive integers around p is a multiple of three which cannot be p itself. Since eight and three are coprime, 24 divides, and 24 is exactly maximal. https://www.youtube-nocookie.com/embed/Jvg7PuWbAxA 54 2026-08-15T06:00:00Z yes Lambdia 2026-08-15T06:00:00.000Z https://lambdia.com/a/the-fly-flies-twenty-miles Twenty Miles of Fly, and the Series Nobody Should Finish https://i.ytimg.com/vi/-qDGL7R7Hts/hqdefault.jpg Two riders twenty-five miles apart close at fifty miles an hour, so a forty mile an hour fly shuttling between them flies exactly twenty miles. The series of shuttle legs gives the same twenty, with a first leg of 100/7 and a round-trip ratio of 1/21, but no finite number of legs ever reaches it: after eighty legs the exact total is twenty minus about 2.6e-52. The general law is wD/(a+b), and it holds identically in the rider speeds rather than by luck at 20 and 30. https://www.youtube-nocookie.com/embed/-qDGL7R7Hts 44 2026-08-07T12:00:00Z yes Lambdia 2026-08-07T12:00:00.000Z https://lambdia.com/a/five-coins-against-four-is-a-coin-flip Five Coins Against Four Is Exactly a Coin Flip https://i.ytimg.com/vi/IuglTT2w82U/hqdefault.jpg You toss five fair coins, I toss four, and you win on strictly more heads: the answer is exactly 256 of the 512 outcomes. Because you hold one coin more, "not strictly more heads" and "strictly more tails" are the same event, and turning every coin over is a bijection between them. The fifth coin is worth nearly fourteen percentage points over the 93/256 you would have without it, and none of that is an edge. https://www.youtube-nocookie.com/embed/IuglTT2w82U 44 2026-08-07T15:00:00Z yes Lambdia 2026-08-07T15:00:00.000Z https://lambdia.com/a/only-perfect-squares-stay-lit A Hundred Bulbs, 482 Flips, and Ten Survivors https://i.ytimg.com/vi/_vfuQxkS2-o/hqdefault.jpg Person k flips every bulb that is a multiple of k, and after a hundred passes exactly the ten perfect squares are lit. Bulb n is flipped once per divisor, and the pairing d against n/d is fixed-point free unless n is a square, so the parity is decided by algebra rather than by accumulation. The lit fraction is one over the square root of the row, and stopping the process at person 50 inverts the answer to 54 bulbs. https://www.youtube-nocookie.com/embed/_vfuQxkS2-o 45 2026-08-07T18:00:00Z yes Lambdia 2026-08-07T18:00:00.000Z https://lambdia.com/a/833-apples-across-a-thousand-miles 833 Apples, and the Surplus Apple That Rides for Free https://i.ytimg.com/vi/fCbwJrJPzyY/hqdefault.jpg A driver who eats one apple per loaded mile delivers 833 of 3000 across a thousand miles, because the price of a mile is the ceiling of the stock over the truck's capacity: three apples, then two, then one. The continuous optimum is 2500/3, but rounding the switch point down to mile 333 leaves 2001 apples needing three passes and produces a fake 834. A lower bound on loaded traversals proves 833 optimal without a dynamic programme, and the free return legs are the clause that separates 833 from 533. https://www.youtube-nocookie.com/embed/fCbwJrJPzyY 50 2026-08-08T06:00:00Z yes Lambdia 2026-08-08T06:00:00.000Z https://lambdia.com/a/hands-at-three-fifteen-are-7-point-5-degrees Seven and a Half Degrees at a Quarter Past Three https://i.ytimg.com/vi/VfEObVfpPfs/hqdefault.jpg At 3:15 the minute hand is on the 3 and the angle between the hands looks like zero. It is 7.5 degrees, or pi/24 radians, because the hour hand crawls a quarter of the way from the 3 to the 4 while the minute hand travels a full lap. The zero answer is exact for a clock whose hour hand waits on each numeral and jumps, which is not a clock that exists. https://www.youtube-nocookie.com/embed/VfEObVfpPfs 42 2026-08-08T09:00:00Z yes Lambdia 2026-08-08T09:00:00.000Z https://lambdia.com/a/product-of-two-uniforms-beats-half-15-percent Two Random Numbers, One Hyperbola, and 15.3 Percent https://i.ytimg.com/vi/iebJ3pwYYC4/hqdefault.jpg Draw X and Y uniformly from the unit interval and their product beats a half with probability (1 - ln 2)/2, about 15.3 percent. The reflex answer of a quarter counts a condition that is genuinely necessary and treats it as sufficient, which is why 0.8 times 0.6 sits inside the quarter square and still loses. The hyperbola y = 1/(2x) cuts the winners down to a sliver, and one integral measures it. https://www.youtube-nocookie.com/embed/iebJ3pwYYC4 42 2026-08-08T12:00:00Z yes Lambdia 2026-08-08T12:00:00.000Z https://lambdia.com/a/checkerboard-stacks-fill-a-cube A Board of Stacks That Folds Into a Cube https://i.ytimg.com/vi/aUhAi4RjaV4/hqdefault.jpg Stack i + j - 1 cubes on every square of a 20 by 20 board and the total is 8000, which is 20 cubed. Folding the board across the squares that are exactly 20 deep pairs every stack with a mirror stack, and each pair adds to 40, so the average depth is 20. The double sum gets the same answer and is merely slow, and the main diagonal is the fold that proves nothing. https://www.youtube-nocookie.com/embed/aUhAi4RjaV4 51 2026-08-08T15:00:00Z yes Lambdia 2026-08-08T15:00:00.000Z https://lambdia.com/a/rock-overboard-lowers-the-water Why Throwing a Rock Overboard Lowers the Pool https://i.ytimg.com/vi/cSfLhq-jWCU/hqdefault.jpg A boat carrying a dense rock floats in a pool; the rock goes over the side and sinks. The mass inside the pool is unchanged, so the reflex says the level cannot move, but it falls by exactly (d-1)V/A. The article carries the algebra the fifty-second version had no room for, plus the force balance on the sunk rock that shows the floor is where the argument closes. https://www.youtube-nocookie.com/embed/cSfLhq-jWCU 45 2026-08-08T18:00:00Z yes Lambdia 2026-08-08T18:00:00.000Z https://lambdia.com/a/real-probabilities-do-not-price-the-call An 80% Chance of $20 Does Not Make the Option Worth $16 https://i.ytimg.com/vi/5cU4I-_RyEA/hqdefault.jpg A stock at 100 goes to 130 with probability 0.8 or 70 with probability 0.2, rates are zero, and the right to buy at 110 is worth 10 rather than 16. A third of a share funded by borrowing 70/3 reproduces both payoffs and costs 10 today, which prices the option without using a probability anywhere. The general risk-neutral probability (S-d)/(u-d) is a half here only because rates are zero and 100 sits midway between the two outcomes. https://www.youtube-nocookie.com/embed/5cU4I-_RyEA 55 2026-08-09T06:00:00Z yes Lambdia 2026-08-09T06:00:00.000Z https://lambdia.com/a/seven-sixteenths-chance-of-meeting Two People, One Hour, and Seven Chances in Sixteen https://i.ytimg.com/vi/OViBuQgCLv8/hqdefault.jpg Two people arrive at random inside the same hour and each waits fifteen minutes, so the reflex answer is a quarter. Drawing both arrival times as one point in a 60 by 60 square turns the question into an area, and the two corner triangles it leaves out have legs of 45, giving 7/16 rather than 1/4. The general formula n(2T-n)/T squared shows why the first minutes of patience buy the most. https://www.youtube-nocookie.com/embed/OViBuQgCLv8 47 2026-08-09T09:00:00Z yes Lambdia 2026-08-09T09:00:00.000Z https://lambdia.com/a/weakest-player-should-waste-the-turn The Weakest Player Should Waste the Turn https://i.ytimg.com/vi/NIvPJupmu1I/hqdefault.jpg You hit one time in ten, your two opponents three and six, and you shoot first. Firing into the air is worth 965/4736 = 20.376%, which beats removing the strongest player by 0.195 percentage points, because a landed hit drops you into the duel you must enter second at 7/37 rather than first at 10/37. The article carries all three option values, the fixed points they solve, and the single Nash equilibrium that turns the usual assumption into a conclusion. https://www.youtube-nocookie.com/embed/NIvPJupmu1I 48 2026-08-09T12:00:00Z yes Lambdia 2026-08-09T12:00:00.000Z https://lambdia.com/a/mutilated-chessboard-resists-dominoes Sixty-Two Squares, Thirty-One Dominoes, and No Arrangement At All https://i.ytimg.com/vi/kxLgrEIrwVg/hqdefault.jpg Cut two diagonally opposite corners off a chessboard and 62 = 2 x 31 stays true, yet nothing fits. Writing the colour of a square as the sign (-1)^(i+j) turns the argument into arithmetic: every domino sums to zero, the two lost corners both carried +1, and the board left over is 30 against 32. The converse, Gomory's theorem, is the harder half and it goes the other way. https://www.youtube-nocookie.com/embed/kxLgrEIrwVg 43 2026-08-09T15:00:00Z yes Lambdia 2026-08-09T15:00:00.000Z https://lambdia.com/a/infinitely-many-south-east-north-loops South, East, North, and Home from Uncountably Many Places https://i.ytimg.com/vi/sp91kvzB8Q0/hqdefault.jpg The North Pole really is a solution, so the trap is only the words "and nowhere else". A mile north of the parallel whose whole lap measures 1/n of a mile, the eastward mile is n exact revolutions, which puts a starting circle 1.159 miles from the South Pole for one lap, 1.080 for two, 1.053 for three. Every point of every circle works, so the honest count is uncountable rather than infinite. https://www.youtube-nocookie.com/embed/sp91kvzB8Q0 50 2026-08-09T18:00:00Z yes Lambdia 2026-08-09T18:00:00.000Z https://lambdia.com/a/ten-heads-in-a-row-is-a-coin-flip You probably think ten heads means a loaded coin https://i.ytimg.com/vi/aAHpjbtS3Pg/hqdefault.jpg Draw one coin from a thousand, flip ten heads, and the chance it is the two-headed one is 0.5062. Both reflex answers miss, in opposite directions: ninety-nine percent ignores the bag, one in a thousand ignores the flips. Counting patterns gets the exact figure with no Bayes notation at all, and the reason it lands on a coin flip is that 2^10 happens to sit next to the size of the bag. https://www.youtube-nocookie.com/embed/aAHpjbtS3Pg 613 2026-08-03T11:00:00Z yes Lambdia 2026-08-03T11:00:00.000Z https://lambdia.com/a/snail-on-a-ten-foot-pole Why does this snail puzzle break on the last day? https://i.ytimg.com/vi/qJMMBsGDFu4/hqdefault.jpg Three feet up each day, one foot back each night, ten feet to climb. Dividing ten by the net two feet a day gives five days, and it charges the snail for a night it never spends. The dawn heights settle the day, the last climb settles the moment, and the closed form the video had no room to voice is a single ceiling function. https://www.youtube-nocookie.com/embed/qJMMBsGDFu4 565 2026-08-03T08:00:00Z yes Lambdia 2026-08-03T08:00:00.000Z https://lambdia.com/a/one-lone-white-marble-gives-75-percent One move beats the fifty fifty split every time https://i.ytimg.com/vi/Czy36QsGzgw/hqdefault.jpg Fifty white marbles, fifty black, two jars, and a fair coin choosing which jar gets drawn from. The even split gives exactly one half, and so does every other split where the jars are the same size. Isolating a single white marble reaches 74/99, an exchange argument proves nothing beats it, and three quarters turns out to be a ceiling no arrangement ever touches. https://www.youtube-nocookie.com/embed/Czy36QsGzgw 699 2026-08-02T17:00:00Z yes Lambdia 2026-08-02T17:00:00.000Z https://lambdia.com/a/683-chips-for-100-cookies Can you guess how many chips a hundred cookies need? https://i.ytimg.com/vi/LWIArp86p0U/hqdefault.jpg Drop chips at random into dough, cut it into a hundred cookies, and ask how many chips guarantee no bare cookie nine times out of ten. Five hundred chips, five per cookie on average, works about half the time. Inclusion-exclusion pins the answer at 683, a closed form you can solve on a whiteboard agrees, and the coupon collector's mean of 518.7 is the sophisticated wrong answer. https://www.youtube-nocookie.com/embed/LWIArp86p0U 616 2026-08-02T14:00:00Z yes Lambdia 2026-08-02T14:00:00.000Z https://lambdia.com/a/never-respin-the-cylinder You probably think that blank told you nothing https://i.ytimg.com/vi/UfsKaww9BQQ/hqdefault.jpg Six slots in a ring, two of them marked side by side. You land on a blank one and get one move: step forward, or draw a fresh slot at random. Both look like two in six. Stepping is one in four, drawing again is one in three, and the whole gap comes from the fact that the two marks are touching. Pull them apart and the advice reverses. https://www.youtube-nocookie.com/embed/UfsKaww9BQQ 556 2026-08-02T11:00:00Z yes Lambdia 2026-08-02T11:00:00.000Z https://lambdia.com/a/cube-outer-layer-488 Almost everyone gets this cube question wrong https://i.ytimg.com/vi/RwUtDxfaadE/hqdefault.jpg Strip the outer shell off a ten-by-ten-by-ten block of unit cubes and count what falls. The reflex answer, 271, is arithmetic done correctly on the wrong picture: a shell leaves from both opposing faces, so every axis loses two units and not one. Two independent counts land on 488, and the general shell turns out to grow like a surface rather than a volume. https://www.youtube-nocookie.com/embed/RwUtDxfaadE 591 2026-08-02T08:00:00Z yes Lambdia 2026-08-02T08:00:00.000Z https://lambdia.com/a/bayes-theorem-false-positive-paradox Your positive test result. Almost everyone reads it wrong. https://i.ytimg.com/vi/32JruhITzzE/hqdefault.jpg A disease one person in two hundred carries, and a test with no false negatives at all. The reflex answer to a positive result is above ninety percent, and it is wrong by more than a factor of ten. A crowd of a thousand people shows why before the algebra does, and Bayes puts the exact figure at 100/1493. https://www.youtube-nocookie.com/embed/32JruhITzzE 240 2026-07-31 yes Lambdia 2026-07-31T00:00:00.000Z https://lambdia.com/a/zero-volatility-option This Option Looks Worthless. Wall Street Pays $4.76 For It. https://i.ytimg.com/vi/NAgz65xbYx4/hqdefault.jpg A stock with zero volatility, a call struck at the money, and the reflex answer — zero — that fails real trading interviews. One arbitrage argument prices it three different ways, and Black–Scholes agrees on the way out. https://www.youtube-nocookie.com/embed/NAgz65xbYx4 410 2026-07-22 yes Lambdia 2026-07-22T00:00:00.000Z https://lambdia.com/a/x-plus-one-over-x-power-sums 2026-07-21T00:00:00.000Z https://lambdia.com/fr/a/x-plus-one-over-x-power-sums 2026-07-21T00:00:00.000Z https://lambdia.com/a/random-triangle-circle-center Why do random triangles catch the circle's center exactly ¼ of the time? https://i.ytimg.com/vi/SJnMzsgC0X4/hqdefault.jpg Drop three random points on a circle and the triangle contains the center exactly a quarter of the time. Two proofs: an honest average, then two coins wearing a geometry costume. https://www.youtube-nocookie.com/embed/SJnMzsgC0X4 937 2026-07-19 yes Lambdia 2026-07-19T00:00:00.000Z https://lambdia.com/a/support-vector-machines-widest-street Support Vector Machines: find the widest street https://i.ytimg.com/vi/IM-MO_QX8tk/hqdefault.jpg Infinitely many lines separate the same data; only the widest street generalizes. And once it's found, almost none of your points were holding it up. https://www.youtube-nocookie.com/embed/IM-MO_QX8tk 704 2026-07-19 yes Lambdia 2026-07-19T00:00:00.000Z https://lambdia.com/a/every-tournament-has-a-hamiltonian-path Every tournament hides a winning chain — even with no clear champion https://i.ytimg.com/vi/ltwBIjps338/hqdefault.jpg Round-robin results can be pure chaos, cycles everywhere, no champion. A one-move induction still lines everyone up so each player beat the next. https://www.youtube-nocookie.com/embed/ltwBIjps338 591 2026-07-19 yes Lambdia 2026-07-19T00:00:00.000Z https://lambdia.com/a/polynomials-that-map-q-onto-q Which polynomials map ℚ onto ℚ? (deceptively simple, brutally deep) https://i.ytimg.com/vi/XAQdYts7hlk/hqdefault.jpg A line with rational coefficients maps ℚ onto ℚ. Nothing curved ever does. Three filters — interpolation, shape, denominators — leave the full classification. https://www.youtube-nocookie.com/embed/XAQdYts7hlk 1096 2026-07-19 yes Lambdia 2026-07-19T00:00:00.000Z https://lambdia.com/a/ninety-coins-need-five-weighings Ninety coins, one fake. How few weighings can you promise? https://i.ytimg.com/vi/s2Gwg6nLdlY/hqdefault.jpg A counterfeit coin that might be heavy or light, a balance that only reports which side falls, and a hundred dollars a weighing. Counting rules out four; only a construction gets you five. https://www.youtube-nocookie.com/embed/s2Gwg6nLdlY 764 2026-08-01T01:00:00Z yes Lambdia 2026-08-01T01:00:00.000Z https://lambdia.com/a/riemann-vs-lebesgue Every Upper Sum Says One, Every Lower Sum Says Zero, and the Gap Is the Measure of the https://i.ytimg.com/vi/x7xUGIygStY/hqdefault.jpg On the unit interval the indicator of the rationals has upper sum 1 and lower sum 0 for every partition ever written, so the two never meet and Riemann returns nothing at all. Give the k-th rational an interval of width ε/2^k and the whole countable set sits inside a total length of ε, which puts its measure at 0 and its Lebesgue integral at 0. Lebesgue's criterion turns that into the general law — a bounded function on a compact interval is Riemann integrable exactly when its discontinuities have measure zero — which is why Thomae's function, discontinuous on the same dense set, is integrable and this one is not. The trade is not free: sin x over x on the half line has an improper Riemann value of π/2 and no Lebesgue integral at all. https://www.youtube-nocookie.com/embed/x7xUGIygStY 682 2026-08-01T12:00:00Z yes Lambdia 2026-08-01T12:00:00.000Z https://lambdia.com/a/two-children-girl-at-the-door 2026-08-05T06:00:00.000Z https://lambdia.com/ebook https://lambdia.com/ebook/fr https://lambdia.com/ebook/es https://lambdia.com/play https://lambdia.com/live 2026-10-03T13:47:43.241Z https://lambdia.com/quandle 2026-10-03T13:47:43.241Z