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.
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.
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.
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.
One condition, a two-line recurrence, and the fifth power falls out as a clean 123 with no radicals left. Climb the same ladder far enough and the golden ratio and the Lucas numbers are hiding underneath.
A line with rational coefficients maps ℚ onto ℚ. Nothing curved ever does. Three filters — interpolation, shape, denominators — leave the full classification.