Game Theory Basics - Department of Mathematics

1.10. SUMS OF NIM HEAPS 13Theorem 1.10 Let n ≥ 1, and n = 2 a + 2 b + 2 c + ··· , where a > b > c > ··· ≥ 0. Then∗ n ≡ ∗(2 a ) + ∗(2 b ) + ∗(2 c ) + ···. (1.7)We first discuss the implications of this theorem, and then prove it. The right-handside of (1.7) is a sum of games, whereas n itself is represented as a sum of powers of two.Any n is uniquely given as such a sum. This amounts to the binary representation of n,which, if n < 2 a+1 , gives n as the sum of all powers of two 2 a ,2 a−1 ,2 a−2 ,...,2 0 , eachpower multiplied with one or zero. These ones and zeros are then the digits in the binaryrepresentation of n. For example,13 = 8 + 4 + 1 = 1 × 2 3 + 1 × 2 2 + 0 × 2 1 + 1 × 2 0 ,so that 13 in decimal is written as 1101 in binary. Theorem 1.10 uses only the powers oftwo 2 a ,2 b ,2 c ,... that correspond to the digits “one” in the binary representation of n.Equation (1.7) shows that ∗n is equivalent to the game sum of many nim heaps, all ofwhich have a size that is a power of two. Any other nim heap ∗m is also a sum of suchgames, so that ∗n + ∗m is a game sum of several heaps, where equal heaps cancel out inpairs. The remaining heap sizes are all distinct powers of two, which can be added togive the size of the single nim heap ∗k that is equivalent to ∗n + ∗m. As an example, letn = 13 = 8+4+1 and m = 11 = 8+2+1. Then ∗n+∗m ≡ ∗8+∗4+∗1+∗8+∗2+∗1 ≡∗4 + ∗2 ≡ ∗6, which we can also write as 13 ⊕ 11 = 6. In particular, ∗13 + ∗11 + ∗6 is alosing game, which would be very laborious to show without the theorem.One consequence of theorem 1.10 is that the nim sum of two numbers never exceedstheir ordinary sum. Moreover, if both numbers are less than some power of two, then sois their nim sum.Lemma 1.11 Let 0 ≤ p,q < 2 a . Then ∗p+∗q ≡ ∗r where 0 ≤ r < 2 a , that is, r = p⊕q 0. We have q < 2 a .By inductive assumption, ∗q ≡ ∗(2 b ) + ∗(2 c ) + ···, so all we have to prove is that ∗n ≡∗(2 a ) + ∗q in order to show (1.7). We show this using lemma 1.8, that is, by showingthat the options of the games ∗n and ∗(2 a ) + ∗q are all equivalent. The options of ∗n are∗0,∗1,∗2,...,∗(n − 1).