# Generalization of Sprague-Grundy Theorem

In my research on Combinatorial Game Theory, I used a certain theorem that is essentially a generalization of the Sprague-Grundy theorem. Because the result hinges too much on the work of others to be truly considered my own, I tried to find a source for citing it. However, I cannot find any source to cite it from.

Define the disjunctive $k$-sum of $n$ games for $n \geq k$ to be the game in which the $n$ games are played in parallel, with each player being allowed to move in no more than $k$ of the games per turn. We can determine the Sprague-Grundy value of a position in the following way:

1. Find the Sprague-Grundy values of each component game.
2. Write each value in base $k + 1$
3. Now add the values without carrying.

The resultant value is the Sprague-Grundy value of the cumulative position.

This theorem is just Sprague-Grundy using Moore's Nim rather than normal Nim. So how should I cite this? As I said previously, I am hesitant to claim the result because it relies so heavily on Moore's Nim. On the other hand, I have not found it anywhere.

• I didn't notice the difference between Moore's results and this generalization until you pointed it out. I'd just cite both Moore and Sprague-Grundy as I think it's an immediate consequence of these two theorems. – Douglas Zare Jun 23 '15 at 5:44