Counting down to a birthday: The Wknd Puzzle with Dilip D’Souza

How many people would you need in a room, to be sure at least two have the same birthday? And what’s the one factor that could dramatically alter those odds?

One famous story about the great logician Raymond Smullyan (1919-2017) is from the time someone introduced him as “unique”. Smullyan interrupted with: “Forgive me sir, but I happen to be the only person in the universe who is, in fact, not unique.”

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Smullyan was one of the great minds of our time. In his book, A Mixed Bag, he tells another favourite story, of a class he was teaching at Princeton. He had just introduced the students to the famous birthday “paradox” that isn’t really a paradox, which goes like this:

Let’s say you’re at a party with about 23 people present. Time for a trick that never fails to induce gasps if it works out, and there’s a better than even chance that it will work out. Take a bet that at least two people in the party have the same birthday. Indeed, you’re more likely to win than not. (Try it, and the interesting puzzle of working out the mathematics.)

At Princeton, there were only 19 students in his class, so Smullyan didn’t try the trick. “There’s only a small chance,” he told them sadly, “that two of you have the same birthday.”

But someone in the class piped up: “Wait! I’m willing to bet that two people in this class share a birthday.”

“I see what you’re doing, pal!” Smullyan said. “You just happen to know the birthday of someone else in the class and it just happens to be the same as yours.”

The student replied: “Not at all! Apart from mine, I know nobody’s birthday here, I swear.”

Smullyan was persuaded. Convinced he had an excellent chance of winning, he took the student’s bet. But Smullyan lost. Why?

Scroll down for the answer.

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Answer:

Unknown to or unnoticed by Smullyan, the class contained a pair of twins.