Questions tagged [game-theory]

Puzzles that involve strategic decision making, usually involving either competitive or cooperative choices between a number of participants.

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3 votes
2 answers
227 views

Directed Graphs from numbers

Bob and Alice play a game. Bob sends a sequence of positive numbers to Alice and using that information she forms a directed graph. For each number in the sequence, she splits it into two non-empty ...
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3 votes
1 answer
153 views

Trapping the Adventurer: the Value of a Gambling Game

An Adventurer is standing on the first of eight sequential platforms, ready to advance. But a Booby-trapper is positioning a trap to catch them in a game played over several rounds. Each round, the ...
  • 199
7 votes
3 answers
1k views

Beyond earth and countries fighting for land

Consider 2 countries, A,B that have discovered a new planet, the size of Earth. They have decided to split the planet into regions of minimum area $10^3 m^2$ and maximum area $10^5 m^2$, with borders ...
  • 189
14 votes
2 answers
620 views

Who is most likely to win all the chips?

Alice, Bob and Charlie each have 5 chips. Starting from Alice, they take turns to act in the order of Alice, Bob and Charlie. The $n$-th person to act must take a total of $n$ chips from their ...
  • 5,279
7 votes
2 answers
863 views

What number should the first person announce?

Five people each announce a number in turn. The number must be an integer in the range from $1$ to $100$, and different from the ones already announced by others. The winner (or winners) of the game ...
  • 5,279
26 votes
9 answers
3k views

N prisoners locked in cells

N prisoners are locked in N separate cells, numbered 1 to N. The warden visits each cell in order with a bowl. Each prisoner can decide whether to put a single coin in the bowl, depending on its ...
  • 5,279
2 votes
1 answer
158 views

Hitting twice with different choices

This game of two players has public parameters an integer $n\ge2$, and a probability $p$ with $1/n<p\le1$. E.g. $n=4$, $p=1/3$. In the first phase of a game, a player secretly decides $n$ ...
  • 383
5 votes
2 answers
495 views

The Median Game - For Money

Inspired by this interesting puzzle which was quickly solved. Five friends play a simple game with the following rules: Players play consecutively one after the other. Each player must call out a ...
  • 7,252
9 votes
1 answer
547 views

The median game

Five friends play a simple game with the following rules: Players play consecutively one after the other. Each player must call out a whole number between 1 and 10 (inclusive), such that it hasn't ...
1 vote
5 answers
354 views

Two wrongs can make a right

Can you construct two single-player games A and B, such that: Every time you play game A you are guaranteed to lose. Every time you play game B you are guaranteed to lose. Every time you alternate ...
1 vote
1 answer
297 views

Pokemon battle puzzle: Chansey vs Wobbuffet

This is a 1v1 battle between two pokemons: Chansey vs Wobbuffet. Your task is to analyze the battle and predict its outcome. Information about the battle: The battle takes place in Generation III. ...
  • 7,189
3 votes
2 answers
392 views

Three-player Nim

Suppose I'm forced to enter the game of nim with two players who are perfectly logical people. The rules is simple: We play a set number of rounds. In every round, a random number of marbles (between ...
  • 31
4 votes
2 answers
507 views

Catch the angel in less than 7 units of time

The devil has trapped the angel in a regular hexagram of firewalls. The perimeter of the hexagram is 12. The devil starts at the apex of the hexagram. can move at speed $1$ to leave a trajectory of ...
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18 votes
4 answers
1k views

How long can you survive at the devil's playground?

The devil has trapped you in his playground. The devil knows that you can't cross over the burning boundary of his circle, so he allows you to choose a position within the circle before he starts to ...
  • 5,279
8 votes
0 answers
338 views

A Glutton and a Split Check

N, T, A, B are arbitrary constants in this problem. I have only definitively solved the N=1 subproblem, so, if the general problem cannot be solved, I will be satisfied to mark as accepted a solution ...
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-1 votes
1 answer
271 views

The Box of Tic-Tacs

You were kidnapped and had no way of escape. The kidnapper opens the lights, and in front of you is an opened box of Tic-Tacs. He claims that he ate from 0-5 Tic-Tacs, (with 60 from the newly-bought ...
7 votes
2 answers
350 views

Can the fugitive escape?

A fugitive is surrounded by N police officers, with the nearest one at distance 1 away. The fugitive and the officers move alternatively. In a fugitive move, the fugitive can travel no more than a ...
  • 5,279
4 votes
1 answer
224 views

Winning strategy in game

Context: I played this game at one point (and lost) and now I'm wondering whether it was possible to win or not. We have a hexagonal board like this: I'm defining the following terms: Each hexagon ...
  • 143
12 votes
3 answers
853 views

Outmaneuver your opponents in the duel

A, B and C are in a three way duel. Starting with A, they rotate in the order of A-B-C, each firing one shot at a time. They stand close to one another, so that each can kill one of the others or ...
  • 5,279
1 vote
1 answer
213 views

Variation of 100 prisoners [duplicate]

There are 100 prisoners, each prisoner is given a number between 1 and 100 without knowing which number, the number can repeat and not all of the numbers must be given. Each prisoner in his turn can ...
  • 11
18 votes
1 answer
605 views

Lap Theory Optimal Betting Game

Three prisoners are seated at a table. Each of them has a mobile phone on their lap, and they are not allowed to look at anyone else’s phone (and obviously no other form of communication is allowed). ...
  • 6,631
9 votes
2 answers
488 views

Three way duel, which guns to choose?

Alice, Bob and Carole are involved in a game of three way duel. They take turns to shoot in the order of Alice-Bob-Carole-Alice-... until only one surviver is left standing. The rule is very simple: ...
  • 5,279
12 votes
4 answers
778 views

A packing game!

Amy and Ben are playing a game which is suggested by a genie. Amy first chooses $a,b,c\in\mathbb{R}^+$. Then a empty cuboid box with internal measurements $a+b,b+c,c+a$, and infinite supply of cuboid ...
  • 5,542
16 votes
2 answers
704 views

Topological Nim

You and your friend's new favorite game is "topological nim": You take your favorite compact metric space $X$ and a radius $r$. Each player removes an open disk of radius $r$ from the space ...
  • 1,848
10 votes
2 answers
642 views

Buzzword Bingo Boss Battle!

Two bosses of different companies take turns delivering long speeches. Every speech much “claim” at least one unused square by uttering the corresponding buzzword(s) in the table below. Victory is ...
  • 6,631
7 votes
2 answers
446 views

A game of 101 stones—win at (not quite) all costs

In the land of Humilia, a tournament game is played between two players with 101 stones in a pot between them. On each turn, a player may take up to five stones from the pot (and must take at least ...
  • 1,848
7 votes
3 answers
981 views

The 15 Pebbles Game

This is a game for 2 players - Each player uses a different coloured marker or pencil, there are 15 pebbles in total. Players take turns to colour 1, 2 or 3 pebbles (player chooses how many). When all ...
10 votes
3 answers
396 views

Find a good strategy to survive the multiplayer duel

Alice is involved in a duel with $N$ other opponents. She can choose a gun for herself with any hit probability as she wishes. She also prepares guns for her opponents, whose hit probabilities she can ...
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5 votes
3 answers
413 views

Playing the Square Chess

On an ordinary chess board (8x8) without any pieces on it as a starting position, two players will alternately place a pawn on any square (unoccupied) so that the first player that form a square (...
  • 6,606
9 votes
2 answers
298 views

A knight chased by four knights

This is a follow up to A knight chased by three knights Two players are playing a variant of chess on a 11x11 grid. The first player controls a white knight that starts in the centre square. The ...
11 votes
3 answers
816 views

A knight chased by three knights

Two players are playing a variant of chess on a 8x8 grid. The first player controls a white knight that starts in the top-left corner. The second player controls three black knights that start in the ...
6 votes
2 answers
276 views

Simple game of dice - what is the optimal strategy?

The game is pretty simple, whoever reaches 50 first wins, but there are a few important details: 2 players, one die. Players take turns. Rolled values are simply added up If you roll a 6, however, ...
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26 votes
5 answers
6k views

Shortest game of Monopoly

This video claims that the fastest game of monopoly involves 4 turns and 9 rolls of the dice. I am wondering if this is actually the shortest game or can we do better in theory? What is the ...
5 votes
2 answers
364 views

Achieve the best outcome by a tactical statement

This puzzle is taken from Schelling's Game Theory: How to Make Decisions by R.V. Dodge, in which bargaining power is gained, paradoxically, by burning bridges and removing options for oneself. Here it ...
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6 votes
1 answer
301 views

The sadistic chieftain and the lethal truel game

In an adventure to a remote island, you get caught by the xenophobic locals and are about to be executed. But the tribe chieftain, known for his sadistic taste, suddenly changes his mind. "I'll ...
  • 5,279
10 votes
2 answers
322 views

Add a divisor! A game

Let $k$ be a positive integer. Amy and Ben are playing a game, with the number $1$ written on the whiteboard initially. Amy and Ben do the following in order, starting with Amy: Suppose the number on ...
  • 5,542
8 votes
5 answers
597 views

Which gun is Alice's best choice?

Alice, Bob and Charles are involved in a deadly three way duel game. Before the game starts, the referee prepares three guns for the players to choose from. She randomly picks three guns from a very ...
  • 5,279
8 votes
3 answers
434 views

Rock, Paper, Scissors, Anti-Rock

Alice and Bob are both well-known perfect logicians who have the ability to internally randomize and often pass their time by playing rock-paper-scissors. One day, tired of the usual RPS game, Alice ...
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7 votes
1 answer
542 views

Playing blindfolded 2048

Suppose you are playing many 2048 games, all blindfolded. As usual you can press left, right, down, or up to move the tiles. When there are no more moves the game terminates and you are told your ...
1 vote
3 answers
157 views

playing "Coins on a table" against a distracted friend [duplicate]

This puzzle is a follow-up of Coins on a table (warning: the link contains the answer to the original question). You are playing a game against your friend Alice: first Alice places one coin on a ...
  • 5,042
10 votes
1 answer
454 views

Is Oh Tetris Oh! always a draw?

Alice and Bob were playing Oh-Tetris-Oh!, when they decided that every game must be a draw (played intelligently). Can you prove this? (Oh yes, the rules: A $10\times10$ grid, you can pick any ...
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3 votes
2 answers
150 views

How to effectively hide an earth rebooter thingy [The Good Place Season 4 spoilers]

This puzzle is unsolved. I don't know if there is a solution or how to solve it. I'm also not sure what type of puzzle this would be, so if anyone wants to correct the tags, feel free. Imagine you ...
  • 133
6 votes
1 answer
366 views

Russian Roulette (Redux!)

You are challenged to a game of Russian roulette with a six-chamber revolver, but you and your opponent are allowed to change the outcome of the game in different ways. Here's how the game will ...
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8 votes
12 answers
3k views

Time to call the bluff [closed]

You are located outside of two rooms which have a door between them. Each room has one or more computers whose total power is unknown, so they cannot be reliably used as proof-of-work to compare times,...
  • 197
10 votes
1 answer
382 views

Reverse dots and boxes – Guess the last move

Alice and Bob are playing a game of reverse dots and boxes. The players take turns adding one horizontal or vertical line of size 1 in one free spot on the grid (marked with light gray lines ...
  • 68.8k
12 votes
1 answer
725 views

Reverse dots and... cubes? - Snowflake edition

A Reverse dots-and-boxes puzzle inspired by @jafe Albert and Robinson are playing a game of reverse dots and boxes. The players take turns adding one plane in one free spot on the grid (...
8 votes
2 answers
845 views

Reverse dots and boxes, swastika edition

Alistair and Roberta are playing a game of reverse dots and boxes. The players take turns adding one horizontal or vertical line in one free spot on the grid (marked with light gray lines in ...
  • 68.8k
14 votes
2 answers
833 views

Reverse dots and boxes

Alice and Bob are playing a game of reverse dots and boxes. The rules are simple: The players take turns adding one horizontal or vertical line in one free spot on the grid (marked with light ...
  • 68.8k
4 votes
1 answer
281 views

Tactical Domino Placement Game

Alice and Bob take turns to place regular dominoes into a $7\times8$ board. The first player who cannot go loses. Is there a winning strategy for either player? Note that in this version the pips on ...
  • 35.4k
8 votes
1 answer
371 views

No point shuffling, just pick your cards

Alice and Bob are playing a card game with the following rules: Alice selects two piles of 3 cards from a standard 52-card deck and places the piles in front of her, face up. Bob can see these ...
  • 68.8k