How the Ladder Game Works
A ladder draw, also called ghost leg or amidakuji, has trivially simple rules yet produces results nobody can predict. There is a mathematical reason for that. Here is why two people can never land on the same outcome, and how many rungs it takes to mix things properly.
The basic rules
Draw several vertical lines and scatter horizontal rungs between adjacent pairs. Start at the top of one line and travel down; whenever you meet a rung you must cross to the neighbouring line and continue downwards. That single rule connects each starting point to exactly one ending point.
The key phrase is must cross. There is no choice involved, so the result is fully determined once the ladder is drawn. The only thing anyone controls is where the rungs go.
The Japanese name amidakuji is said to come from the radiating lines depicted behind the Amida Buddha, which the fan-shaped early versions resembled.
Why two people never collide
A single rung swaps the positions of two adjacent lines. In mathematics that swap is called a transposition, and stringing transpositions together produces a permutation.
A swap is always reversible: do it twice and you are back where you started. Chain any number of reversible operations together and the whole chain stays reversible. That makes the entire ladder a one-to-one mapping.
That is exactly why two people cannot reach the same outcome. If they could, running the ladder backwards from that outcome would have no defined answer, which is a contradiction. However tangled the ladder looks, the property holds.
How many rungs are enough
With too few rungs, results cluster near their starting positions. With no rungs at all, the first line goes to the first outcome and nothing moves.
To mix thoroughly you need rungs on the order of the square of the number of participants. Five participants want around twenty-five, eight want sixty or more, before every permutation becomes roughly equally likely. In practice, visible bias disappears well before that.
The ladder tool here scales the rung count to the number of participants automatically, so results mix properly without anyone having to think about it.
Choosing order does not matter
Whether the ladder is drawn first and participants then pick starting points, or starting points are assigned before the ladder is drawn, the probabilities are identical. As long as the rungs are placed randomly, no starting position carries an advantage.
Human intuition disagrees. People often believe picking first is better, or that the middle lines are safer. Neither is true.
It also matters that the result already exists. The instant the ladder is complete, every connection is fixed, and nothing about who picks when changes any of them.
Practical use
It fits role and forfeit assignment most naturally. Put the actual outcomes into the result slots: win, lose, cleanup, presenting, and so on.
Keeping the outcomes hidden and revealing them one participant at a time is the most enjoyable format. Showing everything at once kills the tension. The ladder here traces each path visually, which preserves that feeling.
The number of outcomes must equal the number of participants. If your draw is mostly losers, add several identical none entries to make up the count.
Trust in an online ladder
A ladder drawn on paper is verifiable because everyone sees the same picture. Online, the computation is invisible, so trust has to be established another way.
The best approach is showing the complete ladder before revealing any results. Watching a path traced through a structure that is already fixed makes it obvious the outcome was not decided afterwards.
A separate article covers this topic in more depth, including exactly where an organiser could interfere and how to close each gap.
Try it right here
You can try what this article describes without leaving the page.
Entry mode
Ladder board
Enter participant count and outcomes, then create a ladder.
Open the full tool
Full screen, saving results, and the rest of the options all live on the tool page.
FAQ
Do more rungs make it fairer?
Past a certain point, no. Roughly the square of the participant count is enough to mix thoroughly, and adding more barely shifts the distribution. Too few, though, and results cluster near their starting positions.
Is picking last a disadvantage?
No. Every connection is fixed the moment the ladder is complete, so the order in which people pick has no effect. Only the number of unclaimed positions changes.
Can two people get the same outcome?
It is structurally impossible. Rungs are position swaps, and chaining swaps always preserves a one-to-one mapping.
What if I have more participants than outcomes?
They have to match. Add filler entries such as none to balance the count. The tool creates the right number of input boxes once you set the participant count.
Can I save the result as an image?
Yes, and it captures both the ladder shape and the connections, which is useful if anyone queries the outcome later.