The Hardest Games of 24

In elementary school, my class would play a game called 24. The teacher would hold up a card that had 4 numbers on it. Using each number on the card exactly once and only +,-,\div,\times  for operations, the goal was to make the number 24. And do it faster than all the other kids, obviously.


2((2 \times 4)+4)=24

A couple days ago, one of my friends gave us a particularly good one. So I thought I’d share the 5 hardest 24 puzzles! Some other people have already gone through the trouble of figuring out which are the most difficult. Enjoy!

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The Sprague-Grundy Theorem

The overall goal of this post is to prove that every impartial game is equivalent to a game of nim. While this result is often called the Sprague-Grundy theorem, the theorem doesn’t actually say anything about this conclusion. After proving the theorem, I’ll explain why a game’s equivalence to nim immediately follows.

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Nim is a simple game, in which 2 players take turns removing objects from piles until there are no more objects left. This post will define basic terms for combinatorial game theory, detail the rules of nim, and explain how to win the game.

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History of Common Games

Games are the perfect mix of chaos and strategy: they are dynamic, since players’ decisions affect them, but there often exists an optimal way to play. This post discusses

  • why we can solve some games but not others.
  • how we solve these games.
  • different processes developed along the way.

The solvability of tic-tac-toe, set take-away, Connect 4, checkers, and chess will be examined in further detail.

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