Coin based subtraction game I'm having a problem in Game Theory where I am trying to understand how a subtraction game can be interpreted by a coin based game.
From my book: 
The problem I'm having is if I have 9 coins and the subtraction set $ \{\ 1,2,3 \}\ $, say, and 3 of them are heads, let's say positions 5, 6 and 7 are heads $(TTTTHHHTT)$. And I want to subtract 2 from this heap of 18...I'd turn over the 7 and then turn over the 5, leaving me with 6, not 16!
I explain it here: http://www.youtube.com/watch?v=eRqxC2j1Oxg&feature=youtu.be

If you have three heads coins $5$, $6$, $7$, this is equivalent to three piles of size $5$, $6$ and $7$, not a single pile of $18$ !
If you remove $2$ from the $7$-pile, you obtain three piles of size $5$, $5$ and $6$. But any impartial combinatorial game combined with itself is irrelevant (the second player can always copy the moves of the first player). Hence this is really equivalent to a pile of $6$.
