When it comes to Gottfried Leibniz (1646-1716), I do a lot of gushing. I gushed in my book on calculus. I gushed in a three-part series of interviews with the podcast Infinitely Irrational. I even gushed in a personality quiz for Ars Technica. The fellow had a spectacular soul, a spectacular mind, and a truly spectacular wig.
But once upon a time, he made a rather spectacular mistake.
Here’s the question. When you roll two standard dice, which is likelier: a sum of 11, or a sum of 12?
Leibniz, in one of his journals, claims that the two are equally probable, since each can be made only “one” way: 5 + 6, or 6 + 6.
But if you’ve played enough board games, or run the calculation yourself, you know that 11 is twice as common. To see why, paint the dice two colors. Only one combination (red 6 + blue 6) gives 12, but two combinations (red 6 + blue 5, or red 5 + blue 6) yield 11.
We all make mistakes. Every few years, I walk face-first into a street sign. My point is not that I’ve overrated Leibniz; it’s that I may have underrated combinatorics. Mathematicians view it as a field of subtle and tricky problems, each distinct from the others, with few broad rules to fall back on. Only patience and experience will reveal its guiding principles.
Remember Leibniz’s blunder next time you make what feels like a “stupid” math error. Simple surfaces belie deeper challenges.