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/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 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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/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/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/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/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/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/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 2026-08-14T08:14:21.531Z https://lambdia.com/quandle 2026-08-14T08:14:21.531Z