Two friends were bored and decided to play a game... a mathematical game with a paper bag!

a photo of a paper bag

Problem statement

John and Mary have a bag full of integer numbers. In fact, the bag has \(10^{10^{10}}\) integers, each written on a plastic card, and the sum of all the \(10^{10^{10}}\) integers in the bag is \(0\). In turns, Mary and John are going to play with the bag by doing the following:

  • Picking two cards from the bag with numbers \(a\) and \(b\) and removing them from the bag;
  • Inserting a new card in the bag with the number \(a^3 + b^3\).

Is there any initial number configuration and/or set of moves for which it is possible that, after \(10^{10^{10}} - 1\) moves, the only card in the bag has the number \(73\)?

Give it some thought... and most important of all, try it for real! Let me know how it went in the comment section below ;)

Hint: the answer is "no". Can you show why?

Hint: look for an invariant of the game! That is, find a property of the game that does not change when Mary and John play it.

Solution

You can read the solution here to compare with your own solution.

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