Ladder Game Probability: Is It Really Random?

The Korean ladder game (사다리타기/Sadari-tagi) is a popular way to make random decisions. But how random is it?

How the Ladder Game Works

1. Draw vertical lines (one per participant)

2. Add horizontal rungs randomly between adjacent lines

3. Each player starts at the top of a line and follows it down

4. When hitting a rung, move horizontally to the next line, then continue down

5. Where you end up determines your outcome

The Mathematics

Key Property: It is a Permutation

Every ladder game produces a permutation — each starting position maps to exactly one ending position, and every ending position is reached by exactly one starting position. No one gets left out.

Is Every Outcome Equally Likely?

With truly random rung placement: approximately yes, but not perfectly.

With very few rungs: Some permutations are more likely than others
With many rungs: The distribution approaches uniformity
The "identity" permutation (everyone stays in their lane) is always possible but becomes less likely with more rungs

Number of Possible Outcomes

For N players, there are N! (N factorial) possible arrangements:

2 players: 2 outcomes
3 players: 6 outcomes
4 players: 24 outcomes
5 players: 120 outcomes

Practical Fairness

In real-world use with a reasonable number of rungs:

2 players: Perfectly fair (always 50/50)
3 players: Very nearly fair with 5+ rungs
4+ players: Fair enough for practical purposes with sufficient rungs

Fun Facts

The ladder game is mathematically equivalent to a sequence of adjacent transpositions
It is used in Korea for everything from deciding who pays for lunch to assigning dormitory rooms
You can prove that ANY permutation can be achieved with the right rung arrangement

Tips for Fairness

1. Add at least 2× as many rungs as there are players

2. Distribute rungs randomly across the full height

3. Have someone who is not playing draw the rungs

4. Use a digital ladder game for true randomness

Use our ladder game tool for perfectly random results.