Every ladder game rests on an unspoken promise no one doubts: "no two people ever land on the same spot." That promise isn't luck — it holds mathematically, always.
Why? The answer is in what a single rung really is. In a ladder, one rung is exactly the same as "swapping the positions of two neighboring vertical lines." In mathematics, an operation that swaps just two elements is called a transposition. Each time your path meets a rung on the way down, the two strands being traced swap places once.
The key property of a swap is that it is reversible: swap two positions, swap the same two again, and you're back where you started. A reversible operation is a one-to-one correspondence (bijection). And chaining bijections together still yields a bijection.
A whole ladder is nothing more than such swaps stacked from top to bottom. Therefore there is always a perfect one-to-one correspondence between all the starting points and all the ending points: (1) two different starts never reach the same end, and (2) every end is assigned to exactly one person. That is precisely what "fair" means here.
"Never overlapping" and "any result is possible" are different claims — and the latter is also true. A theorem says any permutation can be built by repeating swaps of neighboring elements. The sorting method known as bubble sort works on exactly this principle. Turned around: with enough rungs, a ladder can represent any ordering at all, even one that completely reverses the columns.
One caveat. When rungs are placed at random, not every possible ordering appears with exactly equal probability. With few rungs, moves are small, so nearby positions are more likely. But as you add rungs (raise the complexity), the distribution spreads out more evenly. The crucial point: uniformity depends on complexity, yet "no collisions" is guaranteed whether there is one rung or a hundred. Behind a 700-year-old game lies remarkably solid math.
🪜 Raise the complexity and see for yourself →