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.
I don’t know if I’d smoke him at Monopoly, but I definitely would at Can’t Stop (and he’d be confused at the shape of the board).
Of course the lesson here is that painting the dice changes the odds. If you play a game of chance with Ben Orlin, make sure he doesn’t paint the dice when you’re not looking. 🙂
It helps to have two dice in different colors (white with black pips, red with white pips, say) to see that red 5 + white 6 is not the same as white 5 + red 6.
Ben, as a fledgling data scientist, while of course I understand your solution, I sometimes test these out by writing a simulation program in R. By the way I once gave a talk entitled, “Those who can, do – those who can’t, use computer simulation.”
That’s because you can tell one die from another. In quantum theory, you can’t tell one electron from another so if you have two electrons there are are three possible spin states, up-up, down-down and up-down. Since you can’t tell one from another, you can’t tell up-down from down-up and neither can Mother Nature.
The obvious takeaway is that one can alter one’s luck by simply painting the dice. Make sure Ben Orlin doesn’t sneakily paint the dice before a game of chance.
The obvious takeaway is that one can alter one’s luck by simply painting the dice. Make sure Ben Orlin doesn’t sneakily paint the dice before a game of chance.!!