# Find the Grundy number of the initial position and make the first move in a winning strategy for the following game

Find the Grundy number of the initial position and make the first move in a winning strategy for the following game:

In a pile there are two red balls, four green balls, four blue balls, and six white balls. A player can take a (non-zero) number of balls of same color: one red, or one or two green, or one, two, or three blue, or up to four white balls.

How do I do this? I've only ever done Nim games with multiple piles; never have I had one pile with multiple types of objects in one pile.

$$\begin{array}{ccc} &&&\circ\\ &&&\circ\\ &\color{green}\bullet&\color{blue}\bullet&\circ\\ &\color{green}\bullet&\color{blue}\bullet&\circ\\ \color{red}\bullet&\color{green}\bullet&\color{blue}\bullet&\circ\\ \color{red}\bullet&\color{green}\bullet&\color{blue}\bullet&\circ \end{array}$$
The red pile is one-heap takeaway with a limit of $1$ ball per move; the green pile is one-heap takeaway with a limit of $2$ balls per move; the blue pile is one-heap takeaway with a limit of $3$ balls per move; and the white pile is one-heap takeaway with a limit of $4$ balls per move. Your game is in effect the sum of these four simple games.
• So is the initial position for red, green, blue, and white equal to $0, 1, 1, 1$ respectively? And so then the Grundy number of the initial position is eual to $1$? One good first move would be taking one from the green pile, for example? – BrianW May 2 '16 at 21:30
• @BrianW: I get $0,1,0,1$, meaning that there is no winning move. – Brian M. Scott May 2 '16 at 21:39