Games as Numbers

The Game of Hackenstrings

Hackenstrings is a two-player game about capturing tokens. To play, you need a collection of tokens in two colors, which we will assume are Black and White. One player owns the black tokens, the other owns white. Begin by arranging the tokens in rows; there can be any number of rows, any number of tokens in each row, and any mix of black and white tokens in each row that you like. You do not need to use the same number of black and white tokens.

    The goal of Hackenstrings is to make it so that your opponent has no legal move. A move consists of choosing a token of your own color and removing it from the board, along with all tokens to the right of it.


Hackenstrings Arithmetic

Let us call a Hackenstrings position positive if black wins no matter who goes first, negative if white wins no matter who goes first, and zero if the first player to move loses. If we are given two Hackenstrings positions, we can add them together by stacking the rows. Let us give the Hackenstrings position with a single row of n black tokens the value +n.

Really Large and Really Small Games

We have seen how Hackenstrings games can have integer values and certain fractional values. If we allow our rows of tokens to become infinitely long, we can find even more exotic numbers.
Using the Hackenstrings rules, try to find even more infinite or even more infinitessimal games!

Further Reading

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