Tag Difficulty – Hard

Hard brain teasers and puzzles

Prison Keys Strategy

A prison warden was feeling capricious and played a game with the prison keys:

  1. Each prisoner is handed a key to another prisoner’s cell.
  2. Each prisoner will know which other prisoner was initially given the key to their cell (but does not know whose key they were handed).
  3. Each day, when all prisoners are out of their cells and no one is watching, each prisoner is allowed to place keys in another prisoner’s cell.
  4. Each night, each prisoner can collect any keys placed in their cell.
  5. The prisoners can summon the warden when they’re sure everyone has their own key – but if they are wrong, they’re immediately executed.
  6. The prisoners can discuss a strategy beforehand but cannot communicate in any way after keys are handed out.

What is the fewest number of days it would take for the prisoners to be sure everyone has their key? What was the strategy to achieve this?

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Sharing Game Theory Puzzle

In this sharing game theory puzzle, 3 friends take turns taking from a jar of 1000 candies to share. For example, the 1st friend could take 500 candies, then the 2nd friend could take 400, and the 3rd friend would take the remaining 100.

No one wants to be seen as greedy, but no one wants to end up with the fewest candies either. As such, their goals are (in order of preference):

  1. Do not end up with the most candies, nor the fewest candies (a tie for most or fewest also fails this condition)
  2. End up with as many candies as possible

All of them are logical, rational, know each other’s goals, but cannot communicate before or during sharing. How many candies should each friend end up with?

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The Impossible Puzzle

This puzzle was coined the “Impossible Puzzle” by Martin Gardner, a famous math and science writer that liked to create and write about math games and puzzles. The puzzle is named as such because it appears to provide insufficient information to solve, but it is in fact solvable! Read on for the puzzle and the (very difficult) solution.


There are two distinct whole numbers greater than 1, we can call them x and y (where y > x). We know the sum of x and y is no more than 100.

Sam and Prada are perfect logicians. Sam (“sum”) is told x + y and Prada (“product”) is told x * y, and both of them know all the information provided so far.

Sam and Prada have this conversation in which they truthfully deduce the numbers:

  1. Sam: I know Prada does not know x and y.
  2. Prada: Well now I know x and y.
  3. Sam: Ah, now I also know x and y.

Can you figure out x and y using this information?

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Truel

A truel is a three-way duel. The three participants take turns firing one shot at whichever opponent they choose, until only one is remaining.

Allison, Ben, and Chase are in a truel, and have varying degrees of accuracy: Allison has a 50% chance of hitting her intended target, Ben has a 80% chance, and Chase has a 100% chance. Allison gets to shoot first, then Ben, then Chase, and repeating in that order until only one person is remaining.

Assume that the accuracy of all participants are publicly known, and everyone is trying to maximize their chances of winning. What is Allison’s optimal strategy, and what is her likelihood of winning under that strategy?

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