A “curious paradox” presented by Raymond Smullyan at the first Gathering for Gardner: Consider two positive integers, *x* and *y*. One is twice as great as the other, but we’re not told which is which.

- If
*x*is greater than*y*, then*x*= 2*y*and the excess of*x*over*y*is equal to*y*. On the other hand, if*y*is greater than*x*, then*x*= 0.5*y*and the excess of*y*over*x*is*y*– 0.5*y*= 0.5*y*. Since*y*is greater than 0.5*y*, then we can say generally that the excess of*x*over*y*, if*x*is greater than*y*, is*greater*than the excess of*y*over*x*, if*y*is greater than*x*. - Let
*d*be the difference between*x*and*y*. This is the same as saying that it’s equal to the lesser of the two. Generally, then, the excess of*x*over*y*, if*x*is greater than*y*, is*equal*to the excess of*y*over*x*, if*y*is greater than*x*.

The two conclusions contradict one another, so something is amiss. But what?