Ulam and Solitaire

Little did he know that this new form of probability theory would soon play a major role in one of the most important developments of the 20th century. On the morning of the 16th of July, 1945, the United States detonated The Gadget, the world's first nuclear bomb. The six kilogram plutonium bomb created an explosion that was equivalent to nearly 25,000 tons of TNT. This was the culmination of the top secret Manhattan Project, a three-year long effort by some of the smartest people alive, including people like J. Robert Oppenheimer, John von Neumann, and a little known mathematician named Stanislaw Ulam.

- [Derek] Even after the war ended, Ulam continued trying to figure out how neutrons behave inside a nuclear bomb. Now, a nuclear bomb works something like this. Say you have a core of uranium-235, then when a neutron hits a U-235 nucleus, the nucleus splits releasing energy and, crucially, two or three more neutrons. If, on average, those new neutrons go on to hit and split more than one other U-235 nucleus, you get a runaway chain reaction, so you have a nuclear bomb.

But uranium-235, the fissile fuel needed for the bombs was really hard to get. So one of the key questions was just how much of it do you need to build a bomb? And this is why Ulam wanted to understand how the neutrons behave. - [Casper] But then in January of 1946, everything came to a halt. Ulam was struck by a sudden and severe case of encephalitis, an inflammation of the brain, that nearly killed him. His recovery was long and slow, with Ulam spending most of his time in beds.

To pass the time, he played a simple card game, Solitaire. But as he played countless games, winning some, losing others, one question kept nagging at him, what are the chances that a randomly-shuffled game of Solitaire could be won? It was a deceivingly difficult problem to solve. Ulam played with all 52 cards where each arrangement created a unique game, so the total number of possible games was 52 factorial, or about eight times 10 to 67.

So solving this analytically was hopeless. But then Ulam had a flash of insight, what if I just play hundreds of games and count how many could be won? That would give him some sort of statistical approximation of the answer.