Is the Ladder Game Really Fair?

Every rung swaps two players

Imagine walking down the ladder one row at a time. When there is a rung at a given height, the two players on either side of it swap places. Everyone else just keeps going down. A whole ladder is nothing more than a series of swaps between neighbours.

That is why the result is always one-to-one: swapping never puts two players in the same spot. Mathematicians call such a mapping a permutation, and with enough well-placed rungs a ladder can produce any permutation at all.

So where does bias come from?

The problem appears when there are only a few rungs. Each rung moves a player by one column, so a player starting on the far left needs to meet at least one rung fewer than the number of players in a row to reach the far right. With sparse rungs most players finish close to where they started.

When we simulated random ten-player ladders with ten rows, a player starting in position 1 practically never reached result 10, while the result directly below them came up far more often than average. Anyone who knows this can place the prize where it is least likely to be reached, and the draw is no longer fair even though it looks random.

Would more rungs help?

Adding many more rows and rungs does reduce the bias, but it never disappears completely, and the ladder becomes hard to draw and hard to follow. Hand-drawn rungs also tend to cluster, so they mix less than the numbers suggest.

Our approach: shuffle the results first

This site works the other way round. It first decides who gets which result with a perfectly even shuffle, and then draws a ladder that leads exactly to that outcome.

• Assigning results: players and results are paired with a Fisher–Yates shuffle driven by your browser’s cryptographic random number generator (crypto.getRandomValues). Every possible pairing is equally likely.
• Top of the ladder: random rungs are drawn so it looks like a real ladder.
• Bottom of the ladder: rungs that swap neighbours are added step by step until everyone lands on their assigned result.

We checked that it is even

We generated 40,000 ladders each for 2, 3, 5 and 10 players. With 10 players, the chance of every player getting every result fell between 9.5% and 10.4%, right around the expected 10%. The small remaining spread is ordinary sampling noise.

So it does not matter which line you pick or where the prize is written, and choosing first gives no advantage.

Tips for a fair game

• After filling in the results, press Shuffle results to mix their order, so whoever typed them cannot pick a favourable spot.
• A new ladder is drawn when you press Ready, so nobody can study the ladder in advance.
• When several people are involved, share your screen while you play so everyone can see the process.