• i. Tower of Hanoi: recursive solution in $2^n-1$ moves, its optimality, the iterative ctz rule, and Frame–Stewart for 4 pegs.
• ii. Water Jug: $c$ is measurable iff $\gcd (a,b)\mid c\le \max (a,b)$; the fewest pourings by BFS over states.
• iii. Wolf Goat Cabbage: the river-crossing classic in seven crossings, found as a shortest path over safe states.
• iv. Bridge Crossing: pair the two slowest: 17 minutes for $\{1,2,5,10\}$, and the optimal rule for $n$ people.
• v. 100 Prisoners: the cycle-following strategy frees everyone with probability about $31\%$.
• vi. Monty Hall: switching wins with probability $\frac{2}{3}$, and the variants where it does not help.
• vii. Secretary Problem: optimal stopping: reject the first $n/e$ candidates, then take the first record; success $1/e$.
• viii. Poisoned Wine: binary encoding: $\lceil {\log }_{2}n\rceil$ tasters find one poisoned bottle among $n$.
• ix. Celebrity Problem: find the person everyone knows and who knows nobody in at most $3(n-1)$ queries.
• x. Puzzle Index by Technique: named puzzles and recreational problems grouped by the technique that solves them.
• xi. Josephus Problem: survivor recurrence, $O(k\log n)$ lap jumps, and the $k=2$ closed form.
• xii. Counterfeit Coin and 12 Coins: the heavy coin in $\lceil {\log }_{3}n\rceil$ weighings; the 12-coin problem and the $(3^w-3)/2$ bound.
• xiii. Egg Dropping: floors coverable with $e$ eggs and $d$ drops, $F(e,d)={\sum }_i\left(\genfrac{}{}{0ex}{}{d}{i}\right)$, and binary search on drops.
• xiv. Lights Out: a GF(2) linear system; light chasing and the minimum number of presses.
• xv. Prisoners and Hats: the parity strategy that saves $n-1$ prisoners, and its variants.
• xvi. River Crossings: missionaries and cannibals and other crossings by BFS over states, with a table of instances.
• xvii. Knight’s Tour and N Queens: counting $n$-queens solutions with bitmasks and Warnsdorff knight tours.
• xviii. Coupon Collector: about $n{H}_n\approx n\ln n$ boxes to collect all $n$ coupons.
• xix. Birthday Problem: 23 people share a birthday with probability over $50\%$; the same bound for hash collisions.
• xx. Gambler’s Ruin: probability of reaching $N$ before $0$, fair and biased, and the expected duration $i(N-i)$.
• xxi. Two Envelopes Paradox: why “always switch” fails, Broome’s infinite-mean prior, and Cover’s randomized threshold.
• xxii. Boy or Girl and Bertrand’s Box: conditional-probability traps: $\frac{1}{3}$ vs $\frac{1}{2}$, the Tuesday boy $\frac{13}{27}$, Bertrand’s box, three prisoners.
• xxiii. St. Petersburg Paradox: an infinite-expectation game, resolved by finite banks, log utility and Feller’s $n{\log }_{2}n$ law.
• xxiv. Buffon’s Needle: crossing probability $\frac{2\ell }{\pi d}$, long needles, Buffon’s noodle, Barbier’s theorem, estimating $\pi$.
• xxv. Ballot Problem: A stays strictly ahead throughout the count with probability $\frac{a-b}{a+b}$ (reflection).
• xxvi. Stable Marriage: Gale–Shapley deferred acceptance: a stable matching always exists and is proposer-optimal.
• xxvii. Blue-Eyed Islanders and Muddy Children: all $k$ blue-eyed islanders leave on night $k$; common knowledge in a possible-worlds model.
• xxviii. Pirates and Gold: backward induction: the senior pirate buys the cheapest votes ($98,0,1,0,1$).
• xxix. 100 Doors (Lockers): exactly the perfect-square doors stay open (they have an odd number of divisors).
• xxx. Rope Burning and Hourglasses: 45 minutes with two ropes, 9 minutes with 4- and 7-minute hourglasses, and exhaustive search over events.
• xxxi. Bachet’s Weights: powers of 2 for one pan, powers of 3 for both pans (balanced ternary): 4 weights cover $\mathrm{1..40}$.
• xxxii. Chicken McNugget (Frobenius Coin Problem): the largest amount not of the form $xa+yb$ is $ab-a-b$, and $\frac{(a-1)(b-1)}{2}$ amounts are impossible.
• xxxiii. Bachet’s Game and the 21 Game: take $\mathrm{1..}k$ stones: the losing positions are the multiples of $k+1$.
• xxxiv. Wythoff’s Game: losing positions $(\lfloor k\phi \rfloor ,\lfloor k{\phi }^2\rfloor )$ via Beatty sequences.
• xxxv. Chomp and Hex: strategy stealing proves first-player wins; Gale’s proof that Hex has no draws.
• xxxvi. Seven Bridges of Königsberg: Euler’s theorem: an Euler walk exists iff 0 or 2 vertices have odd degree.
• xxxvii. Mutilated Chessboard and Tilings: colouring arguments, Gomory’s theorem, $1\times k$ strips, and counting domino tilings.
• xxxviii. Fifteen Puzzle: solvability by permutation parity, $(RC)!/2$ positions, and optimal solving with IDA*.
• xxxix. Rubik’s Cube and Permutation Puzzles: the cube group order $4.3\cdot 10^{19}$ by Schreier–Sims, its three invariants, and God’s number.
• xl. Peg Solitaire: the rule-of-three invariant, the five possible final holes, and a memoized solver.
• xli. Sudoku: bitmask backtracking with minimum remaining values, the exact cover view, and the known counts.
• xlii. Magic Squares: Siamese, doubly even and LUX constructions for every $n\ne 2$, with the magic constant.
• xliii. Latin Squares and Euler’s 36 Officers: orthogonal Latin squares from finite fields and products; no orthogonal pair of order 6.
• xliv. Kirkman’s Schoolgirl Problem: Steiner triple systems (Bose, Skolem) and a 7-day schedule for 15 girls.
• xlv. Party Problem (Ramsey R(3,3) = 6): among 6 people there are 3 mutual friends or strangers; $R(3,3)=6$ and other Ramsey numbers.
• xlvi. Cake Cutting and Fair Division: cut and choose, Dubins–Spanier and Even–Paz (proportional), Selfridge–Conway (envy-free for 3).
• xlvii. Pancake Sorting: sorting by prefix reversals in at most $2n-3$ flips, and the known bounds.
• xlviii. Collatz (3n + 1): the open $3x+1$ conjecture and computing step counts for every start up to $N$.
• xlix. Wheat and Chessboard: $2^{64}-1$ grains, exponential growth, and 64-bit overflow pitfalls.
• l. Fibonacci Rabbits: the rabbit recurrence, fast doubling for $F_n$, and Pisano periodicity.
• li. Ant on a Rubber Rope: the ant always arrives because the harmonic series diverges, after about $e^{L/v}$ seconds.
• lii. Sum and Product Puzzle and Cheryl’s Birthday: knowledge puzzles solved by filtering candidate sets: Freudenthal’s sum and product, Cheryl’s birthday.
• liii. Zebra Puzzle (Einstein’s Riddle): Einstein’s riddle as a permutation CSP solved by backtracking: the Japanese owns the zebra.
• liv. Knights and Knaves: truth-tellers and liars as boolean formulas; the embedded question and the three gods.
• lv. 100 Prisoners and a Light Bulb: the counter strategy and its expected number of days.
• lvi. Bulgarian Solitaire: piles converge to the staircase $\mathrm{1..}k$ within $k^2-k$ moves; an energy argument on diagonals.
• lvii. Chameleons: the mod-3 invariant decides when all chameleons can become one colour.
• lviii. Gold Chain: one cut pays a 7-day worker; $k$ cuts pay up to $(k+1)2^{k+1}-1$ days.
• lix. Bags of Coins in One Weighing: one weighing finds the light bag: take $i$ coins from bag $i$.
• lx. Tromino Tiling: divide and conquer tiles a $2^k\times 2^k$ board minus any cell with L-trominoes.
• lxi. Coins in a Row: interval DP for the optimal difference, and the parity strategy on an even row.
• lxii. Instant Insanity: four coloured cubes solved with two edge-disjoint 2-regular subgraphs.
• lxiii. Jeep Problem: fuel depots reach $\sum \frac{1}{2k-1}$ one way and $\frac{H_n}{2}$ round trip, and nothing better.
• lxiv. Monkey and Coconuts: the smallest pile is $n^{n+1}-n+1$ coconuts (15621 for 5 sailors).
• lxv. Icosian Game: Hamiltonian cycles on the dodecahedron (30 of them); Hamilton’s game; Petersen has none.
• lxvi. 24 Game and Countdown: exhaustive search over pairs with exact fractions; Countdown is the same search.
• lxvii. Chinese Rings (Baguenaudier): the Gray-code puzzle: $\lfloor 2^{n+1}/3\rfloor$ moves remove $n$ rings.
• lxviii. Gas Station Circuit: a start that completes the circuit exists iff total fuel covers total cost; one pass finds it.
• lxix. Socks, Handshakes and Pigeonhole Puzzles: socks, handshakes, divisible blocks, Erdős–Szekeres, Dirichlet: choosing the boxes.
• lxx. Fitch Cheney Five-Card Trick: naming a hidden card from the order of four; the largest deck is $n!+n-1$.