Two rides, two curves: what Limbo and Crash actually charge

Two functions, printed in full

Most casinos describe their odds. We can do something better, because both of these games are short enough to quote entirely. Here is every line of maths that decides a Limbo round:

And here is Crash, which is not the same function:

Two different shapes. Write u = h / 2⁵² so both are in the same terms, and the difference is plain: Limbo produces 0.99 / (1 − u), Crash produces (100 − u) / (100(1 − u)). The 0.99 in Limbo is the house edge, sitting in the numerator as a flat discount. Crash has no such constant — its edge is hiding in the shape.

What each one costs

Turn each function into the only two numbers a player needs: how often a target is reached, and what a bet on that target returns on average.

For Limbo, the chance of reaching a multiplier x is 0.99 / x, so a bet paying x returns x × 0.99/x — exactly 0.99, whatever x you chose. For Crash the chance that the curve *reaches* x is 99 / (100x − 1), one hundredth more generous — but a cash-out at x is paid only when the bust point is *strictly above* x, and the chance of that is 99 / (100x), which is 0.99 / x again. The return is x × 0.99/x = 0.99, the same flat price.

Correction, 2026-09-19. The first version of this table had a Crash return column of 99x/(100x − 1), from 99.99% at ×1.01 down to 99%, computed from the “curve reaches it” chance. The engine pays a target only when the bust point is strictly above it (settleCrash: auto < crashPoint100), which is one hundredth less likely, and that hundredth makes the return exactly 99.00% at every target. The limbo cap ×10,000 pays; in Crash the cap is the bust itself, so the highest target that can pay is ×9,999.99.

So the honest summary is: both games charge one price everywhere, 1% of every unit staked. The functions differ, the shapes differ, and the price does not. What Crash’s formula gives with one hand — a curve that reaches a multiplier a hundredth more often than Limbo’s — the strict settlement takes back with the other, and the two products land on the same 99%. Crash odds, priced prices every cash-out target and the bust distribution in full.

Why two shapes at all

The answer is not that one is better designed. It is that they are commitments to different things.

Limbo settles per player, per bet. Your seed pair and your nonce produce your number, so the natural construction is “take a fair curve and multiply it by 0.99” — an edge you can point at. That flat 0.99 is why the return is identical at every target, which in turn is why Limbo can honestly say the choice of target is a pure variance decision and nothing else.

Crash settles one number for a whole room. Everybody in the round rides the same curve to the same bust, so the draw cannot depend on any individual’s seed. It comes instead from a hash chain — each round’s link hashes to the previous one, with a public salt — so the entire sequence of future rounds was fixed before the first bet was ever placed, and can be checked backwards forever.

The (100 − u)/(100(1 − u)) shape is the standard construction for a chain-derived bust point, and its edge emerges from the arithmetic rather than from a constant someone chose. That is why its return varies with the target while Limbo’s does not — and why comparing the two required deriving both rather than reading a number off a page.

The cap, and what it costs you

Both functions clamp at the same ceiling: 1,000,000 in hundredths, or ×10,000. Limbo’s uncapped curve would keep going; the clamp cuts the tail off both.

What does the cap actually take? For Limbo, the chance of the curve wanting to exceed ×10,000 is 0.99/10,000 — about 1 in 10,101 rounds. In those rounds you are paid ×10,000 instead of whatever the curve produced. The expected cost of the clamp is therefore small but not zero, and it is the one place where Limbo’s otherwise perfectly flat 99% is not quite the whole story.

We point that out rather than round it away, because “exactly 99% at every target” is the kind of clean claim that deserves its own footnote. It is exactly 99% at every target you can actually select; the ceiling is where the model stops.