Where should you stand?
The contest
Imagine you and nine other people are lined up in front of a glass bridge. It has 10 rows, and each row has two glass tiles side by side. In every row, one tile is safe and the other breaks the moment you step on it. Fall through and you're out. The first person to make it all the way across wins a hypothetical $1,000,000, and nobody else gets anything.
Players cross one at a time. Everyone waiting behind gets to watch, so they learn which tiles hold and which ones break. You get to choose your place in line. Which spot gives you the best chance of being the first one across?
The two ends of the line
If you go first, you know nothing. You have to guess all 10 rows correctly, and each guess is 50/50:
(1/2)10 = 1 in 1,024
That's about a 0.1% chance.
If you go last, nearly every row will be solved by the time it's your turn. But you only get a turn if all nine people in front of you fell, and that almost never happens. Someone ahead of you has very likely crossed already. So the best spot is somewhere in the middle. The question is where.
Every crossing is 10 coin flips
Assume everyone remembers every tile they've seen. Then each row is guessed exactly once, by whoever is at the front of the line when it comes up. If they guess right, they move on to the next row. If they guess wrong, they fall, but now everyone behind them knows the safe tile in that row, and the next player walks straight to the first unknown row.
So however it plays out, a crossing is exactly 10 guesses, one per row, each a coin flip. That gives 210 = 1,024 equally likely ways the whole thing can go. That's the bottom of the fraction.
Counting the ways you win
Each wrong guess knocks out one player. If you stand in spot k, you win when exactly k − 1 of the 10 guesses are wrong. You need k − 1 so that everyone in front of you has fallen, and not one more, because the next wrong guess would be you falling through.
The number of ways to pick which k − 1 rows got a wrong guess is "10 choose k − 1," written C(10, k − 1). That's the top of the fraction. For spot 6, you need exactly 5 wrong guesses out of 10, and there are 252 ways to choose those 5 rows:
C(10, 5) ÷ 210 = 252 ÷ 1,024 ≈ 24.6%
The answer: spot 6
Work it out for every spot and you get this:
- Spot 1: 0.1%
- Spot 2: 1.0%
- Spot 3: 4.4%
- Spot 4: 11.7%
- Spot 5: 20.5%
- Spot 6: 24.6%
- Spot 7: 20.5%
- Spot 8: 11.7%
- Spot 9: 4.4%
- Spot 10: 1.0%
Going first is the worst. The odds climb to a peak at spot 6, then fall again, because the further back you stand, the more likely it is that someone ahead of you already made it. Spot 6 has nearly a 1 in 4 chance, so in this hypothetical it would be worth about $246,000.
The peak makes sense once you see the coin flips. Out of 10 flips, the most likely number of wrong guesses is 5, right in the middle. You want to stand just behind the fifth person to fall. There's a neat symmetry too: going first needs zero wrong guesses, and nobody crossing at all needs ten. Each happens in exactly one of the 1,024 ways, so they're equally likely.
A longer bridge with more players keeps the same shape. The best spot is always just past the middle, but the winners spread across more spots, so even the best one gets worse. With 18 rows and 18 players it's spot 10 at 18.5%. With 50 of each it's spot 26 at 11.2%.
Try it yourself
Pick a spot in the game above and watch the line cross. You can choose your own tiles or let them be picked for you. One crossing can't show you 24.6%, so after a run, try "Run 1,000 bridges" and see how often each spot actually wins.