All Problems | AxiomIQ

All Problems, filterable by difficulty and progress
#	Title	Topic	Difficulty	Status
01	100 Prisoners, 100 Boxes Each prisoner may open 50 of 100 boxes to find their own number. Random guessing succeeds with probability (1/2)^100; the optimal strategy does astronomically better.	Logic Puzzles	7.0/10	Unsolved
02	All in a Semicircle Three points are chosen uniformly at random on a circle. What is the probability all three lie within some semicircle?	Probability	6.0/10	Unsolved
03	Amoeba Extinction Each minute an amoeba dies, stays the same, splits in two, or splits in three, each with probability 1/4. What is the probability the population dies out?	Probability	4.0/10	Unsolved
04	Ant on a Cube An ant walks randomly along the edges of a cube. How many moves does it expect to need to reach the opposite vertex?	Probability	5.0/10	Unsolved
05	Arranging MISSISSIPPI How many distinct arrangements are there of the letters in MISSISSIPPI?	Combinatorics	3.0/10	Unsolved
06	Battle of the Sexes A couple prefers being together but disagrees on the venue. In the mixed equilibrium, how often does each attend their own favorite?	Game Theory	8.2/10	Unsolved
07	Benford's Law In naturally occurring datasets, what fraction of numbers start with the digit 1?	Statistics	5.0/10	Unsolved
08	Bertrand Competition Two identical firms set prices and the cheaper one takes the whole market. Where do prices end up?	Game Theory	6.0/10	Unsolved
09	Bertrand's Chord A random chord is drawn by picking two points on a circle. What is the probability it is longer than a side of the inscribed equilateral triangle?	Probability	7.0/10	Unsolved
10	Bessel's Correction A sample consists of 2, 4, and 6. What is the sample variance, and why divide by n − 1?	Statistics	5.0/10	Unsolved
11	Birthday Paradox How many people need to be in a room before there is at least a 50% chance that two of them share a birthday?	Combinatorics	4.0/10	Unsolved
12	Buffon's Needle Drop a needle onto a floor ruled with lines exactly one needle-length apart. What is the probability it crosses a line?	Probability	5.0/10	Unsolved
13	Burning Ropes Two ropes each burn for 60 minutes, but unevenly. Measure exactly 45 minutes: how few times must you put flame to a rope end?	Logic Puzzles	7.0/10	Unsolved
14	Chebyshev's Guarantee For any distribution whatsoever, what fraction of observations must lie within 2 standard deviations of the mean?	Statistics	4.9/10	Unsolved
15	Cheryl's Birthday Albert knows the month, Bernard knows the day, and three cryptic statements pin down the date. When is Cheryl's birthday?	Logic Puzzles	7.0/10	Unsolved
16	Chomp Players eat chunks of a chocolate bar whose bottom-left square is poisoned. Who wins with optimal play, and why is the proof famous?	Game Theory	6.0/10	Unsolved
17	Counting Surjections How many functions from a 5-element set onto a 3-element set hit every element of the codomain?	Combinatorics	2.6/10	Unsolved
18	Coupon Collector There are 6 distinct coupons, each equally likely in a cereal box. How many boxes do you expect to buy to collect all 6?	Probability	4.0/10	Unsolved
19	Cournot Duopoly Two firms simultaneously choose quantities with price P = 120 − Q and zero costs. What does each produce in equilibrium?	Game Theory	7.0/10	Unsolved
20	De Méré's Wager What is the probability of rolling at least one six in four rolls of a fair die, the bet that helped launch probability theory?	Probability	4.0/10	Unsolved
21	Derangements n letters are placed randomly into n addressed envelopes. As n → ∞, what does the probability that no letter ends up in its correct envelope approach?	Combinatorics	5.0/10	Unsolved
22	Distinct Faces A fair die is rolled six times. How many distinct faces do you expect to see?	Probability	4.0/10	Unsolved
23	Eight Rooks In how many ways can 8 rooks be placed on a chessboard so that no two attack each other?	Combinatorics	3.0/10	Unsolved
24	First Heads Wins Alice and Bob alternate flipping a fair coin, Alice first; whoever flips heads first wins. What is the probability Alice wins?	Probability	5.0/10	Unsolved
25	First-Price Auction with Uniform Values Two bidders with uniform private values bid in a first-price auction. What fraction of their value does each bid in equilibrium?	Game Theory	9.4/10	Unsolved
26	Full House What is the probability a random 5-card poker hand is a full house?	Probability	3.0/10	Unsolved
27	Gambler's Ruin A gambler starts with $10 and bets $1 on a fair coin flip each round, stopping at $0 or $20. What is the probability of reaching $20?	Probability	6.0/10	Unsolved
28	Grid Paths How many right/up paths cross an 8 × 8 grid from corner to corner?	Combinatorics	3.0/10	Unsolved
29	Grim Trigger In an infinitely repeated Prisoner's Dilemma, how patient must players be for grim-trigger cooperation to hold?	Game Theory	5.0/10	Unsolved
30	Guaranteed Pair Sum How many numbers must you pick from {1, ..., 100} to guarantee two of them sum to 101?	Combinatorics	4.0/10	Unsolved
31	Halving the Standard Error Your sample of size n gives a standard error SE. How big a sample cuts the standard error in half?	Statistics	5.0/10	Unsolved
32	Hawk-Dove Hawks fight over a resource at a cost; Doves share and flee. With what probability is Hawk played in the evolutionarily stable strategy?	Game Theory	7.0/10	Unsolved
33	Making Change In how many ways can you make 25 cents from pennies, nickels, and dimes?	Combinatorics	4.0/10	Unsolved
34	Matching Pennies Match pennies: Player 1 wins $1 on a match, Player 2 wins $1 on a mismatch. What is the value of this game?	Game Theory	6.0/10	Unsolved
35	Max of Two Dice Roll two fair dice. What is the expected value of the larger of the two values?	Probability	4.0/10	Unsolved
36	Maximum of Uniforms Four values are drawn uniformly from [0, 1]. What is the expected value of the largest one?	Statistics	5.0/10	Unsolved
37	MLE for the Uniform You observe n draws from Uniform(0, θ) with θ unknown. What is the maximum likelihood estimator of θ?	Statistics	6.0/10	Unsolved
38	Monty Hall, Generalized You pick 1 of 4 doors. The host opens 2 losing doors. What is the probability of winning if you switch?	Probability	6.9/10	Unsolved
39	Moser's Circle Join six points on a circle with all chords. The region counts 1, 2, 4, 8, 16 suggest 32 next; what is it really?	Combinatorics	5.0/10	Unsolved
40	Nim 3-4-5 Nim with piles of 3, 4, and 5: the first player wins with optimal play. How many stones does the unique winning first move remove?	Game Theory	7.0/10	Unsolved
41	No Consecutive Elements How many subsets of {1, ..., 10} contain no two consecutive integers?	Combinatorics	4.0/10	Unsolved
42	No Two Heads Adjacent A fair coin is flipped 10 times. What is the probability no two heads land consecutively?	Probability	5.8/10	Unsolved
43	One Hundred Hats 100 prisoners in a line, each guessing their own hat color. How many can be guaranteed to survive with the right strategy?	Logic Puzzles	7.0/10	Unsolved
44	Optimal Dice Stopping Roll a die up to three times; after each roll, keep the value or roll again. What is the expected payout under optimal play?	Probability	7.0/10	Unsolved
45	Partitioning a Set In how many ways can a 5-element set be split into any number of nonempty unlabeled blocks?	Combinatorics	7.0/10	Unsolved
46	Paths Below the Diagonal How many lattice paths from (0,0) to (10,10) never rise above the diagonal? The answer is a famous sequence in disguise.	Combinatorics	3.0/10	Unsolved
47	Penney's Game Two players pick sequences of three coin flips. The player whose sequence appears first wins. How many distinct sequences can Player 2 choose to guarantee a winning edge over any sequence Player 1 picks?	Game Theory	6.0/10	Unsolved
48	Polygon Diagonals How many diagonals does a convex 12-sided polygon have?	Combinatorics	3.0/10	Unsolved
49	Praise and Punishment Praised cadets get worse; berated cadets improve. The instructor credits punishment. What is really going on?	Statistics	6.0/10	Unsolved
50	Prisoners and the Lightbulb 100 prisoners, one light switch, and a fatal declaration to time perfectly. How does the designated counter know when everyone has visited?	Logic Puzzles	7.7/10	Unsolved
51	Pólya's Urn Draw a ball, return it with another of the same color, repeat. What is the probability the third draw is red?	Probability	8.0/10	Unsolved
52	Race to 100 Players alternately add 1–10 to a running total; whoever says exactly 100 wins. What is the winning first move?	Game Theory	6.0/10	Unsolved
53	Random Walk Returns A symmetric random walk on the integers starts at 0. What is the expected number of steps to return to 0?	Probability	9.8/10	Unsolved
54	Roll Until Six Roll a fair die until a six appears, then stop. What is the expected sum of all your rolls?	Probability	4.1/10	Unsolved
55	Seven Before Eight Two dice are rolled repeatedly. What is the probability a sum of 7 shows up before a sum of 8?	Probability	3.0/10	Unsolved
56	Shooting Stars The chance of seeing a shooting star in an hour is 0.84. What is the chance of seeing one in 10 minutes?	Probability	6.0/10	Unsolved
57	Simpson's Paradox Treatment A beats B on small stones AND on large stones. Can B still win overall when the groups are combined?	Statistics	6.0/10	Unsolved
58	Sixty Heads A fair coin is flipped 100 times. Roughly how likely are 60 or more heads?	Statistics	5.0/10	Unsolved
59	Solve the Matrix A zero-sum game with payoff matrix [[3, −1], [−2, 4]]. What is the value of the game?	Game Theory	5.0/10	Unsolved
60	St. Petersburg Paradox A casino flips a coin repeatedly. You win 2^n dollars where n is the flip on which heads first appears. What is the expected value of this game?	Game Theory	5.0/10	Unsolved
61	Stackelberg Leadership Same market as Cournot, but one firm commits to its quantity first. What does the leader produce?	Game Theory	5.0/10	Unsolved
62	Stag Hunt Hunt the stag together for the big payoff, or play it safe with a hare? Find the mixed-equilibrium probability of hunting stag.	Game Theory	6.1/10	Unsolved
63	Stars and Bars How many nonnegative integer solutions does x₁ + x₂ + x₃ + x₄ = 10 have?	Combinatorics	3.6/10	Unsolved
64	The Ballot Problem Candidate A gets 60 votes, Candidate B gets 40. What is the probability A was strictly ahead throughout the entire count?	Combinatorics	4.5/10	Unsolved
65	The Blue-Eyed Islanders A visitor tells 100 blue-eyed perfect logicians something they all already "know." On which night does everything change?	Logic Puzzles	8.6/10	Unsolved
66	The Bridge at Night Four people crossing at speeds 1, 2, 5, and 10 minutes share one flashlight, two at a time. What is the fastest total crossing?	Logic Puzzles	5.0/10	Unsolved
67	The Broken Stick A stick is broken at two uniformly random points. What's the probability the three pieces form a triangle?	Probability	5.0/10	Unsolved
68	The Camel and the Bananas A camel must move 3000 bananas across 1000 km, carrying 1000 at a time and eating 1 per km. How many bananas can arrive?	Logic Puzzles	6.0/10	Unsolved
69	The Census Taker Three children's ages multiply to 36 and sum to the house number next door, yet that isn't enough. One more clue settles it.	Logic Puzzles	5.4/10	Unsolved
70	The Centipede Game Two players could grow a pot over 100 rounds by passing. On which round does backward induction say the game ends?	Game Theory	8.0/10	Unsolved
71	The Die Hard Jugs An unmarked 3-gallon jug, an unmarked 5-gallon jug, unlimited water. How few steps to exactly 4 gallons?	Logic Puzzles	6.0/10	Unsolved
72	The Fly and the Trains A fly zigzags at 75 mph between two trains closing at 100 mph from 100 miles apart. How far does it fly?	Logic Puzzles	6.0/10	Unsolved
73	The Fork in the Road One road leads to safety, one to doom, and the guard is either a truth-teller or a liar. How many yes/no questions do you need?	Logic Puzzles	7.0/10	Unsolved
74	The German Tank Problem You capture four tanks with serial numbers 19, 40, 42, and 60. How many tanks does the enemy have?	Statistics	7.0/10	Unsolved
75	The Handshake Count At a party of 20 people, everyone shakes hands with everyone else exactly once. How many handshakes?	Combinatorics	3.0/10	Unsolved
76	The Hundred Lockers Person k toggles every kth locker, for k = 1 to 100. After all the passes, how many lockers are open?	Logic Puzzles	5.0/10	Unsolved
77	The Josephus Survivor 100 people stand in a circle and every second person is eliminated until one remains. Which position survives?	Combinatorics	5.0/10	Unsolved
78	The Lost Boarding Pass 100 passengers, 100 assigned seats, and one lost ticket. What is the probability the last passenger ends up in their own seat?	Probability	9.0/10	Unsolved
79	The Mislabeled Jars Three jars (apples, oranges, mixed) and every label is wrong. How few fruit draws guarantee correct relabeling?	Logic Puzzles	6.0/10	Unsolved
80	The Missing Bullet Holes WWII bombers returned riddled with holes on the wings but not the engines. Where did Abraham Wald say to put the armor?	Statistics	6.0/10	Unsolved
81	The Mutilated Chessboard Remove two opposite corners from a chessboard. Can 31 dominoes tile the remaining 62 squares?	Combinatorics	5.0/10	Unsolved
82	The Peeking Problem An analyst checks an A/B test's p-value daily and stops the first time p < 0.05. Does the 5% guarantee survive?	Statistics	7.0/10	Unsolved
83	The Pirate Gold Problem Five perfectly rational pirates divide 100 gold coins. The most senior proposes a split; a strict majority rejection means he dies. How many coins does the senior pirate keep?	Game Theory	7.0/10	Unsolved
84	The Prisoner's Dilemma Two suspects are interrogated separately with the classic payoffs. How many years does each serve in the unique Nash equilibrium?	Game Theory	7.0/10	Unsolved
85	The Rare Disease Test A 99%-accurate test for a disease affecting 1% of people comes back positive. What is the chance you actually have it?	Statistics	4.0/10	Unsolved
86	The Rational Ultimatum A proposer splits $100; the responder can accept or leave both with nothing. What does a purely rational proposer offer?	Game Theory	6.0/10	Unsolved
87	The Rendezvous Two friends each arrive at a random time between noon and 1 pm and wait 15 minutes. What is the probability they meet?	Probability	5.0/10	Unsolved
88	The Round Table In how many distinct ways can 8 people sit around a circular table, counting rotations as the same?	Combinatorics	4.0/10	Unsolved
89	The Second-Price Auction The highest bidder wins but pays the second-highest bid. What is the optimal bid, no matter what anyone else does?	Game Theory	6.0/10	Unsolved
90	The Secretary Problem You interview n candidates one by one and must decide immediately after each interview. What is the probability the optimal stopping rule selects the best candidate?	Probability	5.0/10	Unsolved
91	The Sock Drawer Two socks drawn at random from a drawer of red and black socks are both red with probability exactly 1/2. What is the smallest possible drawer?	Probability	5.0/10	Unsolved
92	The Sum and the Product S knows the sum, P knows the product, and four cryptic statements identify two secret numbers. Freudenthal's "Impossible Puzzle."	Logic Puzzles	8.0/10	Unsolved
93	The Taxi Cab Problem 85% of cabs are Green, and a witness who is right 80% of the time says the cab was Blue. How likely was it actually Blue?	Statistics	5.0/10	Unsolved
94	The Truel Three shooters with hit rates 1/3, 2/3, and 1 duel in turns, weakest first. What should the weakest shooter do first?	Game Theory	6.5/10	Unsolved
95	The Twelve Balls Twelve identical-looking balls, one counterfeit of unknown weight, and a balance scale. How few weighings always suffice?	Logic Puzzles	7.0/10	Unsolved
96	The Two Coins One fair coin, one two-headed coin. You pick one at random and flip heads three times. What is the chance it's the trick coin?	Statistics	5.0/10	Unsolved
97	The Waiting Time Paradox Buses average one per 10 minutes and you arrive at a random time. Why is your expected wait not 5 minutes?	Statistics	7.0/10	Unsolved
98	Three Switches, One Bulb Three switches downstairs, one incandescent bulb upstairs. How many inspection trips do you need to match switch to bulb?	Logic Puzzles	6.0/10	Unsolved
99	Threes and Fives How many integers from 1 to 1000 are divisible by 3 or 5?	Combinatorics	2.2/10	Unsolved
100	Tiling with Dominoes How many ways can a 2 × 10 board be tiled with 1 × 2 dominoes?	Combinatorics	3.0/10	Unsolved
101	Trailing Zeros How many zeros does 100! end in?	Combinatorics	3.0/10	Unsolved
102	Traveler's Dilemma Two travelers claim a value from $2 to $100 for lost luggage, with a $2 bonus for the lower claim. What do rational players claim?	Game Theory	7.0/10	Unsolved
103	Twenty-Five Horses 25 horses, a 5-horse track, no timer. How many races to guarantee finding the fastest three?	Logic Puzzles	6.0/10	Unsolved
104	Two Children A family has two children and you learn at least one is a boy. What is the probability both are boys?	Probability	5.0/10	Unsolved
105	Two Eggs, 100 Floors With only two identical eggs and 100 floors, how many drops guarantee you find the critical floor?	Logic Puzzles	6.0/10	Unsolved
106	Two-Color Necklaces How many distinct necklaces of 6 black-or-white beads exist, counting rotations as the same?	Combinatorics	3.3/10	Unsolved
107	Two-Thirds of the Average Everyone picks a number in [0, 100]; closest to two-thirds of the average wins. What do perfectly rational players pick?	Game Theory	7.3/10	Unsolved
108	Variance of Two Dice What is the variance of the sum of two independent fair dice?	Statistics	3.0/10	Unsolved
109	Waiting for Double Heads Flip a fair coin repeatedly. How many flips do you expect to need before seeing two heads in a row?	Probability	4.0/10	Unsolved
110	What a 95% Confidence Interval Means Is it correct to say "there is a 95% probability the true mean lies in this interval"?	Statistics	6.0/10	Unsolved
111	What a p-value Means A test yields p = 0.03. Is that the chance the null is true, the chance the result is luck, or something else?	Statistics	6.0/10	Unsolved
112	Wolf, Goat, and Cabbage A farmer, a one-item boat, and three mutually hungry passengers. What is the minimum number of river crossings?	Logic Puzzles	5.0/10	Unsolved
113	Zero Correlation, Not Independence If two random variables have correlation exactly 0, must they be independent?	Statistics	4.0/10	Unsolved