Who is smarter: a person or an ape? Well, it depends on the task. Consider Ayumu, a young male chimpanzee at Kyoto University who, in a 2007 study, put human memory to shame. Trained on a touch screen, Ayumu could recall a random series of nine numbers, from 1 to 9, and tap them in the right order, even though the numbers had been displayed for just a fraction of a second and then replaced with white squares.

I tried the task myself and could not keep track of more than five numbers—and I was given much more time than the...