Game Description: Odd Scoring

Oops, I previously talked about Odd Scoring without giving a descriptive post. Let me fix that now!

One of my students presented this game in my first-year seminar to the class last semester. Here are the rules.

There are a total of an even number of horizontally-set spaces, with one marker placed at the far-right space. Both players keep their own score, which starts off at 0. Each turn, a player moves the marker 1, 2 or 3 spaces to the left, then adds to their score that number of spaces moved. Once the marker is in the far-left space, the player with an odd score wins.

Some interesting points:

* exactly one player will have an odd score, because there are an odd number of moves the marker will make in total.
* you really only need to keep track of the parity of the players' score. Having 9 and 3 points are the same.
* this is not strictly a combinatorial game; the last player to move may not be the winner!

Even though this game is not quite combinatorial, can it be considered all-small? Whenever one player has a move, the other also has a move. However, in some cases this is a move where the game ends, but the last player to move still loses. Consider the following game state:

_X_ ___ ___ ___ ___ ...

Left: 0 Right: 1

What is the value of this position? That definition may change whether this game is considered all-small or not! This may not be consistent with the definition of "points" considered earlier this week!