White to play and mate in 2

Continue reading “Mate in 2 Chess Puzzle 5”# 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:

- Each prisoner is handed a key to another prisoner’s cell.
- Each prisoner will know which other prisoner was initially given the key to their cell (but does not know whose key they were handed).
- 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.
- Each night, each prisoner can collect any keys placed in their cell.
- The prisoners can summon the warden when they’re sure
*everyone*has their own key – but if they are wrong, they’re immediately executed. - 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?

Continue reading “Prison Keys Strategy”## 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):

- Do not end up with the most candies, nor the fewest candies (a tie for most or fewest also fails this condition)
- 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?

Continue reading “Sharing Game Theory Puzzle”## Converge to an Integer

Given the iterative formula:

*x*_{n+1} = sqrt(*x*_{n}) + *a*

There are some values of *a* for which a positive starting value for *x* results in this formula converging to an integer.

For example, if *a* = 2, you would observe that *x*_{n+1} = sqrt(*x*_{n}) + 2 eventually converges to 4 for any positive starting value.

What form must *a* take in order for this formula to converge to an integer?

## Mate in 4 Chess Puzzle 1

White to play and mate in 4.

Continue reading “Mate in 4 Chess Puzzle 1”## Consecutive Numbers Grid Puzzle

13 | |||

10 | |||

4 | 16 | ||

7 |

The cells in this 4×4 grid puzzle contain the numbers 1-16 each once. There are two rules to the arrangement of the numbers:

- Any two consecutive numbers must share a row/column (1 and 16 should be considered consecutive).
- No row/column can contain three consecutive numbers.

Fill in the remaining numbers.

Continue reading “Consecutive Numbers Grid Puzzle”## 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:

- Sam: I know Prada does not know
*x*and*y*. - Prada: Well now I know
*x*and*y*. - Sam: Ah, now I also know
*x*and*y*.

Can you figure out *x* and *y* using this information?

## Four Lines of Four Coins

Form four lines of four coins by moving exactly 3 of the coins below:

Rules:

- Coins cannot be stacked on top of one another.
- A line of five or more coins is still only considered one line.
- The centers of the coins must be precisely on the same straight line, but can be anywhere on the line.
- Lines can be any length, and can be horizontal, vertical, or diagonal.

## 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?

Continue reading “Truel”## Mate in 2 Chess Puzzle 1

White to play and mate in 2.

Continue reading “Mate in 2 Chess Puzzle 1”