# Putnam Puzzler 2010

**Question:**

Consider a set *S* and a binary operation ***, i.e. for
each *a, b* in *S*, *a*b* is also in *S*.
Assume that *(a*b)*a = b* for all *a, b* in *S*.
Prove that *a*(b*a) = b* for all *a, b* in *S*.

**Solution:**

The hypothesis implies that *((b*a)*b)*(b*a) = b*
for all *a, b* in *S* by replacing *a*
with *b*a*. Hence, *a*(b*a) = b* for all
*a,b* in *S* using *(b*a)*b = a*.