Let’s ease into Monday. Win Smith with the Well Tempered Spreadsheet blog encountered one of those idle little puzzles that captures the imagination and doesn’t let go. It starts as many will with spaghetti.
I’m sure you’re intrigued too. It’s not the case that any old splitting of a strand of spaghetti will give you three pieces you can make into a triangle. You need the lengths of the three pieces to satisfy what’s imaginatively called the Triangle Inequality. That inequality demands the lengths of any two sides have to be greater than the length of the third side. So, suppose we start with spaghetti that’s 12 inches long, and we have it cut into pieces 5, 4, and 3 inches long: that’s fine. If we have it cut into pieces 9, 2, and 1 inches long, we’re stuck.
The Triangle Inequality is often known as the Cauchy Inequality, or the Cauchy-Schwarz Inequality, or the Cauchy-Bunyakovsky-Schwarz Inequality, or if that’s getting too long the CBS Inequality. And some pranksters reorder it to the Cauchy-Schwarz-Bunyakovsky Inequality. The Cauchy (etc) Inequality isn’t quite the same thing as the Triangle Inequality. But it’s an important and useful idea, and the Cauchy (etc) Inequality has the Triangle Inequality as one of its consequences. The name of it so overflows with names because mathematics history is complicated. Augustin-Louis Cauchy published the first proof of it, but for the special case of sums of series. Viktor Bunyakovsky proved a similar version of it for integrals, and has a name that’s so very much fun to say. Hermann Amandus Schwarz first put the proof into its modern form. So who deserves credit for it? Good question. If it influences your decision, know that Cauchy was incredibly prolific and has plenty of things named for him already. He’s got, without exaggeration, about eight hundred papers to his credit. Collecting all his work into a definitive volume took from 1882 to 1974.
Back to the spaghetti. The problem’s a fun one and if you follow the Twitter link above you’ll see the gritty work of mathematicians reasoning out the problem. As with every probability problem ever, the challenge is defining exactly what you’re looking for the probability of. This we call finding the “sample space”, the set of all the possible outcomes and how likely each of them are. Subtle changes in how you think of the problem will change whether you are right.
Smith cleans things up a bit, but preserves the essence of how the answer worked out. The answer that looks most likely correct was developed partly by reasoning and partly by numerical simulation. Numerical simulation is a great blessing for probability problems. Often the easiest way to figure out how likely something is will be trying it. But this does require working out the sample space, and what parts of the sample space represent what you’re interested in. With the information the numerical simulation provided, Smith was able to go back and find an analytic, purely reason-based, answer that looks plausible.