Indistinguishability quotient

The nimbers that occur in the normal play of such positions are *0, *1, *2, and *3.

Nimber	Outcome	Positions with that nimber
${\displaystyle \ast 0}$	P	Even number of heaps of size 1 and even number of heaps of size 2
${\displaystyle \ast 1}$	N	Odd number of heaps of size 1 and even number of heaps of size 2
${\displaystyle \ast 2}$	N	Even number of heaps of size 1 and odd number of heaps of size 2
${\displaystyle \ast 3}$	N	Odd number of heaps of size 1 and odd number of heaps of size 2

These four nim values combine according to the Klein four-group:

${\displaystyle {\begin{array}{c|ccc}+&\ast 0&\ast 1&\ast 2&\ast 3\\\hline \ast 0&\ast 0&\ast 1&\ast 2&\ast 3\\\ast 1&\ast 1&\ast 0&\ast 3&\ast 2\\\ast 2&\ast 2&\ast 3&\ast 0&\ast 1\\\ast 3&\ast 3&\ast 2&\ast 1&\ast 0\end{array}}}$

The Klein four-group is also defined by the commutative group presentation

${\displaystyle \mathrm {V} =\langle a,b\mid a^{2}=b^{2}=1\rangle }$.

The elements of ${\displaystyle \mathrm {V} =\{1,a,b,ab\}}$ can be thought of as in one-to-correspondence with the nim values ${\displaystyle \{\ast 0,\ast 1,\ast 2,\ast 3\}}$ that occur in the play of this simplified Nim game; they combine exactly in the same way.

So far, this formal introduction of the Klein four-group has added nothing new to the conventional analysis of 1- and 2-pile Nim using nimbers and nim-addition. Instead, we have merely recast the theory into a multiplicative form.

The advantage of the multiplicative form is that it allows us to write down a similar solution for the misere quotient of Nim played with heaps of size one and two only.

We introduce the commutative monoid presentation

${\displaystyle \mathrm {M} =\langle a,b\mid a^{2}=1,\ b^{3}=b\rangle .}$

whose six elements are ${\displaystyle \{1,a,b,ab,b^{2},ab^{2}\}.}$

Misere quotient value	Outcome	Positions in that class
1	N	Even number of heaps of size 1 and no heaps of size 2
a	P	Odd number of heaps of size 1 and no heaps of size 2
b	N	Even number of heaps of size 1 and odd number of heaps of size 2
ab	N	Odd number of heaps of size 1 and odd number of heaps of size 2
${\displaystyle b^{2}}$	P	Even number of heaps of size 1 and even number (greater than zero) of heaps of size 2
${\displaystyle ab^{2}}$	N	Odd number of heaps of size 1 and even number (greater than zero) of heaps of size 2

The solution to the correct play of misere Nim was already fully described by Bouton in 1902.[1] In the final sentence of that paper, Bouton writes that in misere Nim, "the safe combinations are the same as before, except that an odd number of piles, each containing one, is now safe [ie, is an P-position], while an even number of ones is not safe [ie, is an N-position]." The misere quotient formulation above is easily seen to be equivalent in the case of Nim played with heaps of size one and two only.