Handshake puzzles and Birthday bets

Warning: use these tools wisely, they can keep nieces and nephews busy for hours

What is the sum of all the numbers from 10 to 1? That’s difficult.

One repeated theme – because it works! – is reframing. The same information but a new presentation changes our understanding.

What is the sum of these numbers: 10, 9, 1, 8, 2, 7, 3, 6, 4, 5? That’s easier. Reframe again, this time visually.

Each what is the sum question can be framed as a triangle. But reframe again, first to staircases, then as pairs.

Doesn’t this feel like magic? The universe presents this little nugget (What is the sum of all the numbers from 10 to 1?) and rather than slog through we skip over. Reframing numbers into shapes changes our tool from addition to multiplication. Magical. We went from brute force to clever pairing to formulaic: (n*(n+1))/2.

Another question: In a group, how many handshakes must occur for everyone to shake everyone else’s hand?

Two people have one handshake. Five have ten connections.

Like triangle numbers, it’s almost the same math! Rather than counting the dots, we want the connections. Rather than (n*(n+1))/2, the formula is (n*(n-1))/2.

It’s not only handshakes to count but games in a round robin, cables to connect computers, and shared birthdays.

In a group of 31 people, what are the chances any two people share a birthday? Thirty-one is a good number because it frames our thinking. “That’s like one month, so one-in-twelve”.


The chances are closer to three-in-four thanks to our connections.

There’s a 99.7% (364/365) chance two people have different birthdays, (.997)1 (connection). The chance of five people having different birthdays, (.997)10, is 97%. Even the chance ten people have different birthdays, (.997)45, is 87%.

But keep going. Thirty-one people have 465 connections and a 25% chance of differing birthdays.

Every day on Twitter, the joke goes, someone is the main character – and you don’t want it to be you. Something is always happening because of this birthday/games/handshake structure. It’s easier not to get wrapped up in “this headline” knowing there will always be headlines.

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