Why Ghost Leg Is Fair: The No-Collision Rule and a Hidden Bias

Draw the rungs of a ghost leg however you like and, oddly, everyone still lands in a different slot. It looks like luck but it's a mathematical certainty. That doesn't automatically mean the odds are fair, though.

Why don't paths collide?

A rung swaps the people on its two neighboring lines. Whoever is coming down the left line moves right, and whoever is on the right line moves left. A swap can never bring two people onto one line.

Everyone starts on a different line, and however many rungs follow, all that ever happens is one swap of two people at a time. So everyone also finishes on a different line. Mathematicians call a matching like this a permutation.

So does it matter which line you pick?

Here's the catch. Without a large number of rungs, people tend to finish near where they started. Moving sideways requires meeting a rung, and traveling far requires that to happen several times in the same direction.

Picture five lines with only a few rungs. To get from the far left to the far right, you must pass through at least four rungs in just the right order. That is far less likely than landing somewhere close to your starting point.

So on a paper ladder where the X is written down first, simply avoiding the line straight above it can give you a small edge, even without tracing a single rung.

Two ways to make it fair

First, place the X at random and keep it hidden. Whatever the paths look like, if the X is equally likely to be in any slot, every starting point has exactly the same chance of reaching it. This is what Ghost Leg on this site does. Each starting point leads to one specific slot, and every slot is equally likely to hold the X.

The same holds with more than one X. With n players and k X slots, everyone is caught with a probability of k/n.

Second, draw a lot of rungs. More rungs mix the finishing positions more evenly, at the cost of a messier ladder. On paper, a good trick is to let someone add a few extra rungs after everyone has chosen a name.

In short

• The structure of a ladder always sends people to different slots.
• The ladder's shape alone doesn't mix finishing positions evenly.
• A hidden, randomly placed X makes the odds equal whatever the ladder looks like.

Next time you draw one on paper, fold over the X and ask someone to add a few more rungs.