Difficulty – Hard

Hard brain teasers and puzzles

Sicherman Dice

Sicherman dice are a particular pair of six-sided dice that, when rolled, produce sums with the same probability distribution as a pair of standard six-sided dice. In other words, these dice do not have the arrangement of 1, 2, 3, 4, 5, and 6 on their six sides, but when rolled together, still produce the same distribution of sums:

  • One way to roll a 2
  • Two ways to roll a 3
  • Three ways to roll a 4
  • Four ways to roll a 5
  • Five ways to roll a 6
  • Six ways to roll a 7
  • Five ways to roll a 8
  • Four ways to roll a 9
  • Three ways to roll a 10
  • Two ways to roll a 11
  • One way to roll a 12

Sicherman dice are the only alternative arrangement of six-sided dice with positive integers that produce the same probability distribution as a pair of standard six-sided dice. Can you figure out what numbers belong on Sicherman dice?

Hint: One of the dice has no numbers greater than 4.

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Who Scored How Many Points

Lalo, Tyson, and Michael played a number of games of pick-up basketball. At the end, their combined points across all the games were:

  • Lalo: 20
  • Tyson: 10
  • Michael: 9

They noticed that in every game, one of them scored x points, one of them scored y points, and one of them scored z points, where x > y > z and all three are distinct positive integers.

If Tyson got the highest score in the first game, who got the second highest score in the second game?

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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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Converge to an Integer

Given the iterative formula:

xn+1 = sqrt(xn) + 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 xn+1 = sqrt(xn) + 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?

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