Is the Ladder Game Really Fair? A Start-Position × Outcome-Option Matrix Analysis

Time to cash in the cliffhanger from last time — "the ladder game (사다리타기) actually isn't fair." When I posted the first version, a reader left a sharp comment: "But it depends on the situation, right? Would the result really be the same with 1 winner and 4 losers as with 5 totally different menu options?" Fair point. So this time I ran a precise analysis with a start-position × outcome-option 5×5 matrix. Across 5 real-world scenarios, I'll work out exactly "who should start where to be favored" 🪜.

1First, the ladder algorithm in one line

How the ladder game decides the outcome is genuinely simple. You take N vertical lines and draw K horizontal rungs at random; whenever a person starting from the top hits a rung, they swap to the adjacent line beside them. That's it.

So a ladder is essentially "a chain of swaps," and the result is 100% determined by the number and positions of the rungs. The more rungs there are, the more the positions get shuffled; the fewer there are, the more likely you are to land near your starting position.

2100,000 simulations — the shocking result

Experiment setup:

• 5 vertical lines (5 participants)
• 5 horizontal rungs placed at random (only between adjacent lines, never two in the same row)
• Mulberry32 PRNG with changing seeds, repeated 100,000 times
• Measured the arrival-position distribution from each starting position

In theory every arrival position should come out to exactly 20.0%. But what's the actual result?

Slot 3 is 23.7%, slot 5 is 16.3%. (23.7 - 16.3) / 16.3 = about 45%. In other words, the middle slot gets picked about 1.45 times as often as the two ends. There's a big hole in the belief that the ladder is "fair."

3Why the middle gets picked more — the mathematical reason

This isn't just a coincidence. Worked out intuitively, here's why:

Reason 1: The middle slot "can receive swaps from both sides"

Slot 1 can only receive a swap from slot 2, even when one happens. Slot 5 can only come from slot 4. But slot 3 can be reached from either slot 2 or slot 4. That's twice the number of entrances.

Reason 2: The ends "only receive a swap from one side"

When there are few rungs, a person starting from an end slot is more likely to experience fewer swaps and arrive where they started. So if your starting position is at an end, your result tends to be at an end too. Someone who starts in the middle gets tossed around this way and that, and in the end tends to get pulled back toward the center.

Reason 3: The fewer the rungs, the bigger the bias

As the number of rungs K grows infinitely large, every position gets swapped evenly and the result eventually converges to a uniform distribution. But the K = 5 to 10 we actually draw is a "partially random" state, so the bias stays right where it is.

4How the bias changes with the number of rungs

I measured how the distribution shifts as the number of rungs goes from 5 to 50:

See the pattern? Once the number of rungs passes 3× the number of people (15), the bias nearly disappears, and at 10× (50) it converges to an almost perfectly uniform distribution. In short:

5The real crux — arrival distribution by starting position (Transition Matrix)

Everything up to here was aggregate statistics. But the ladder's true fairness only becomes clear when you look at "if I start in slot 1, where do I end up." I took the same 100,000 simulations and broke them out by starting position:

This matrix is the ladder's true nature. Four key findings:

• ① Staying in your own slot is the most likely outcome · look at the bold numbers on the diagonal: 1→1 is 36%, 5→5 is 36%. In other words, with few rungs you barely move
• ② Starting at an end produces the biggest bias · someone starting at slot 1 has only a 5% chance of reaching slot 5. In practice, you can't reach half the arrival spots
• ③ Starting in the middle (slot 3) is the most random · spread relatively evenly between 13% and 28%. The slot that leaves it to "true luck"
• ④ Left-right symmetric structure · 1↔5 and 2↔4 are mirror-symmetric. That is, the left end and the right end have the same positional value

65 real-world scenarios — who's favored, and where?

A reader made a great point in the comments — "there's no way the result is the same with 1 winner and 4 losers as with 5 completely different menus." True. Applying the matrix above, here's an analysis of 5 real situations:

📍 Scenario 1 · 5 dinner menus (all different options)

e.g., slot 1 = Korean, slot 2 = Chinese, slot 3 = Japanese, slot 4 = Western, slot 5 = snacks. 5 people randomly placed in starting positions 1 to 5.

• Most frequently picked menu: Japanese (slot 3) — overall arrival rate 23.7%
• Least frequently picked menus: Korean, snacks (slots 1/5) — 16% to 17%
• The trick: put your favorite menu in the middle (slot 3) and start from the middle yourself → 28% chance of getting that menu
• To run it fairly: shuffle the menu positions one more time (random placement)

📍 Scenario 2 · 1 winner, 4 losers (who's buying the coffee — winner in the middle)

e.g., one "winner (= the person who buys)" in slot 3, and the rest all "losers (= those who don't buy)."

The trick: if you can choose your starting position, grab slot 1 or 5 first. An 82% chance of not getting caught (slightly better than the theoretical 80%).

📍 Scenario 3 · 1 winner, 4 losers (winner at an end)

e.g., "winner" in slot 1, the rest losers.

Start from the end opposite the winning position and you have a 95% chance of not getting caught. In other words, "where the winner is + where you start" both drive the result. When you visualize the simulation, the two ends are the most asymmetric.

📍 Scenario 4 · 4 losers, 1 exemption (chore/penalty exemption)

e.g., among 5 people on cleaning duty, only 1 is exempt. Say the "exemption" slot is in the middle (slot 3).

• Starting position with the highest chance of reaching the exemption: slot 3 (28%) ⭐
• Starting position with the lowest chance of reaching the exemption: slots 1/5 (18%)
• The trick: if you know the exemption's position in advance, start from the same line as the exemption
• Fair operation: randomly place both the exemption position and the starting positions before starting the ladder

📍 Scenario 5 · Splitting into two teams (2 vs 3)

There are two ways to arrange it. The difference in outcome is stark:

Method A — adjacent layout: [A, A, B, B, B]

• Starting positions 1 and 2 → combined chance of going to team A about 59%
• Starting positions 4 and 5 → combined chance of going to team B about 76%
• ⚠️ The starting position decides the result roughly 70% of the time → the ladder is almost meaningless

Method B — alternating layout: [A, B, A, B, B]

• From every starting position, the A/B arrival probabilities are nearly even
• ✅ This is true random team allocation
• Pro tip: don't place the team result labels adjacently — arrange them in alternating order

7Two scenarios shown on the actual game screen

I brought the matrix analysis above straight onto the LuckyPlz ladder screen and worked through two scenarios. See how just changing the people and the outcome-option layout shifts the intensity of the asymmetry.

📍 Case A · A housewarming with 5 friends — two chores, three exemptions

Outcome options: Exempt / Dishes / Exempt / Cleaning / Exempt. A situation where 2 of the 5 get caught with a chore.

▲ Case A · 3 exemptions + dishes + cleaning / getting a chore: 41% at the two ends, 45% in the middle

📍 Case B · Friends at a karaoke room — one person for the first song (1 winner, 4 exemptions)

Outcome options: Exempt / First song / Exempt / Exempt / Exempt. A group of 5 at a karaoke room deciding who sings the first song. Watch how starkly the asymmetry shows up when the result piles into a single spot.

▲ Case B · one spot for the first song + four exemptions / a roughly 2.7× gap between the two ends

🆚 The two cases at a glance

Even with the same ladder and the same 5 people, depending on how you arrange the outcome options, the fairness of the result ranges from nearly uniform all the way to 2.7× asymmetric. The more a single outcome like an exemption or a winner piles into one spot, the more overwhelmingly the starting position drives the result — that's the ladder's true nature.

8So how should you use the ladder?

Three conclusions:

Method 1: Lay down enough rungs (number of people × 3 or more)

The textbook approach. With 5 people, 15 or more; with 8, 24 or more. The downside is the screen can get packed and feel a bit visually cluttered.

Method 2: Shuffle the starting positions too

If you're going to draw few rungs, compensate by shuffling the starting positions one more time. Before writing 1 to 5 on the ladder, shuffle the 1-to-5 cards at random and then start.

Method 3: Just use roulette (most recommended)

With roulette, if the slices are equal in area, it's mathematically 100% uniform. No need to worry about "did I lay down enough rungs" like with a ladder. Fast, fair, done.

That said, the ladder has its own charm. Roulette doesn't have "the suspense as you trace your way down." If you're trying to liven up the mood among friends, the ladder is plenty good too. Just lay down enough rungs.

9Practical guide — recommended rungs by number of people

For reference, the LuckyPlz ladder lets you adjust the number of rungs directly. The default is auto-calculated at roughly number of people × 2 to 3, but you can crank it higher with the slider. If fairness matters at your gathering, push the slider all the way once before you start.

10Frequently asked questions

Q. So everything I've decided with a ladder up to now was unfair?

Strictly speaking, yes, but it's a bias on the order of "the middle slot gets picked 50% more often," so it's not a big deal in everyday life. Fine for choosing a dinner menu. But for decisions with serious money or stakes on the line, roulette is recommended.

Q. With 1 winner and 4 losers, if I can pick my starting position, where's best?

If you know the winning position in advance, start from the end opposite the winning position. As we saw in Scenario 3, if the winner is in slot 1, starting from slot 5 gives you a 5% chance of reaching it, which is effectively safe. But if you don't know "where the winner is" and can only pick your starting position, the middle (slot 3) is safe — no matter where the winner is, your chance of reaching it stays between 13% and 28% on average.

Q. When choosing among 5 menus, should I put my favorite in the middle or at an end?

The middle is the answer. As we saw in Scenario 1, the middle arrival rate is 23.7% vs. 16% to 17% at the ends. Put the "menu you really want to eat" in slot 3 and start from the middle (slot 3) too, and your chance of getting that menu rises to 28%. If the other 4 are scattered across slots 1/2/4/5 and you alone start at slot 3, it's even more favorable.

Q. When splitting into two teams, how should I arrange the result labels?

Always use an alternating arrangement. An adjacent layout like [A, A, B, B, B] lets the starting position decide the result 70% of the time, making the ladder practically meaningless. With an alternating layout like [A, B, A, B, B], the A/B probabilities become nearly even from every starting position. See the table in Scenario 5.

Q. So does the LuckyPlz ladder include all these corrections?

The current version includes (1) automatic sufficient rung placement + (2) a starting-position shuffle option. (3) Outcome-option shuffle isn't there yet — the structure places options in the exact order you type them, so you have to enter them at random yourself. We're planning to add an "outcome shuffle" toggle in the next update.

Q. Isn't picking my own starting position an advantage?

Yes. Grabbing the middle position (slot 3 for 5 people) statistically makes you more likely to get caught. That's why the LuckyPlz ladder includes an option to auto-shuffle starting positions too (ON by default). Turn it off and the bias above kicks in as-is.

Q. What happens with 0 rungs?

You just arrive exactly where you started. In other words, no shuffle and the starting order is the result as-is. 100% biased (= no shuffle actually happens at all).

Q. Do ladders on other sites have the same bias?

Yes. It's a property of the ladder algorithm itself, so wherever you run it, the same bias appears whenever there aren't enough rungs. It's not a problem with the site's code — it's a mathematical limitation.

Q. Can you share the simulation code?

It's a JavaScript Mulberry32 PRNG + simple swap logic. I'll clean it up and share a GitHub link in the next post. If you want to run it yourself, hang tight.