Ghost Leg Picker

A ghost leg is a drawing that pairs a list of people with a list of results. You draw one vertical line per person, write a name at the top of each line and a result at the bottom of each line, then scatter short horizontal rungs between neighbouring lines while the results are covered up. To read your answer you start on your own name and walk downwards, and every time a rung touches the line you are standing on you cross it to the neighbouring line and keep going down. Whatever sits at the bottom of the line you arrive on is yours.

It is called Amidakuji in Japan, sadaritagi, meaning ladder climbing, in Korea, and ghost leg in English, which is a translation of the Chinese name gui jiao tu. In all three places it is the ordinary way a group settles who does what: who buys the coffee, who takes which chore, who sits where, who has whom for secret santa. The name Amidakuji comes from the shape of the very old version, in which lines radiated from a centre like the halo of the Amida Buddha and each person pulled one; the ladder of vertical lines and rungs came later and is what everyone draws now.

Why does everyone get a different result?

Because a rung can only ever swap two neighbours. Look at one row of the ladder: every rung in it takes two neighbouring lines and exchanges them, and no line is touched twice in the same row. A row is therefore a rearrangement of the lines with nothing added and nothing lost. Stack the rows and you compose those rearrangements one after another, and a composition of swaps is still a rearrangement, which mathematics calls a permutation.

Two consequences follow, and they are the whole reason the game works. The first is that the path is reversible: start at the bottom of any line and walk upwards under the same rule and you arrive back at exactly one name. The second is that two players can never land on the same result, because that would require one bottom line to be reachable from two different top lines, which reversibility forbids. This is a property of the diagram, not of any particular software, and it holds however many rungs you draw and wherever you draw them.

Is this draw fair?

Fair means every possible pairing is equally likely, and that is a stricter demand than simply scattering rungs. A hand drawn ladder with only three or four rungs is not fair at all: most players end up on the line they started on, because there is nothing to move them. Adding more rungs improves it, and a ladder with a large number of random rungs gets close to even, but close is not the same as even.

This page therefore works the other way round. It first draws a uniformly random pairing of the names with the results, using a shuffle in which each remaining choice is picked with equal probability and the range is taken by rejection rather than by a remainder, so no result is quietly favoured. Only then does it build a ladder that produces exactly that pairing, by laying down a stretch of random rungs and following it with the rungs needed to finish on the drawn answer. Every outcome has the same probability by construction, and the ladder you see is an honest picture of it: trace any line by hand and you will get the answer the page gives.

The randomness itself comes from the browser cryptographic generator, crypto.getRandomValues, and not from Math.random, which is fast but is not required to be unpredictable. What that generator produces is a seed, printed under every ladder. The seed is the only thing the share link carries besides the names and the results, which is why the same link always redraws the identical ladder.

Ghost leg or a spinning wheel?

A wheel answers who. One spin, one winner, and if you need a second winner you spin again and remember to take the first name out. A ghost leg answers who gets what, for everybody, in one draw, and duplicates are impossible rather than merely avoided. If your question is which of six people has to present first, spin a wheel. If it is which of six people takes which of six chores, draw a ladder.