When it comes to Gottfried Leibniz (1646-1716), I bubble a lot. I raved about calculus in my book. I raved about a three-part series of interviews on the Infinitely Irrational podcast. I even raved about on a personality quiz Ars Technica. The guy had a spectacular soul, a spectacular mind, and a really spectacular wig.

But once upon a time it was a pretty spectacular mistake.

Here is the question. If you roll two standard dice, which is more likely: a total of 11 or a total of 12?

Leibniz claims in one of his journals that both are equally likely, since each can only be made “one” way: 5 + 6 or 6 + 6.

But if you’ve played enough board games or done the math yourself, you know that 11 is twice as common. To see why, paint the cubes two colors. Only one combination (red 6 + blue 6) gives 12, but two combinations (red 6 + blue 5 or red 5 + blue 6) give 11.

We all make mistakes. Every few years I go face first on a street sign. My point is not that I overrated Leibniz; It is so that I underestimated the combinatorics. Mathematicians see it as a field of subtle and tricky problems that differ from one another and that few general rules can fall back on. Only patience and experience will reveal the guiding principles.

Remember that the next time Leibniz made a mistake that feels like a “stupid” calculation error. Simple surfaces believe in deeper challenges.

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