The Mathematics of
Grego's Limbo Match
and how merit is weighed over luck
by Grego (Gergely Földvári)
This is a companion to the game — playing explains everything grego.hu/limboAbstract
Grego's Limbo Match is a memory–strategy–lottery game. P blue chips numbered 1 to P sit face-down in a middle stack; their matching reds lie face-down around a dial. A player flips the top blue to read its value, then searches the reds for its pair within a flip limit — or gambles the clover for a blind guess and banks points for unspent flips. A missed search sends the blue to the bottom of the stack, to resurface last with a better chance: the Limbo.
A shuffle of P reds is a permutation, and its loops are its cycles. That one observation is what makes the game mathematical: a loop pattern is a partition of P, the shuffles sharing a pattern are counted by its cycle type, and summed over every partition of P those counts come to P factorial. These three studies follow that thread — through the size of the field a score is set against, through two perfect readers of the same shuffle, and through the question in the subtitle: how a scoring system can weigh merit above luck, and be shown to do so.
The three studies
Part I · 9 pagesThe Maths of the Feast
The feasible tally, the worth of a category, and the merit that ranks a champion. A category is fixed by four dials: P the pairs, 2 to 21; F the flip limit per search; L the limbo come-back chances; and LL, the loop-length window that selects which shuffles can occur. The paper counts what those dials admit — 220,407 categories in all — and works out each one's shuffle pool as an exact sub-sum of P factorial. At P = 10 the window [1, 4] gathers 23 partitions and 632,736 shuffles; a full P = 13 board has 6,227,020,800. A score is therefore placed against a field whose size is known to the last permutation.
Part II · 10 pagesThe Two Readers
Cyrus the cycler follows the loops: each chip he turns names the next chip to turn. Rowen the rower scans the rows in plain order. Two players with perfect memory read one shuffle two different ways — does the way they read change the score, and should the ranking care? The answer is that their expected flips are equal, category by category, both sitting on the harmonic number. A Monte-Carlo run across all 19 world-record categories at 80,000 trials each confirms it: 1961.50 against 1961.45, a difference of 0.05, with the confidence interval straddling zero and wins splitting 49.2 to 48.9 per cent. Under flawless sighted play the two readers finish level.
Part III · 30 pagesAcross the Tiers
Where the depth comes from, and how deep it goes. Among its findings: every flip costs exactly one point, so memory becomes economics; a search costs exactly the length of the loop it walks; the record flip ladder sits on the Golomb–Dickman ratio of 0.62; perfect play clears the board about half the time at every size from 5 to 21 pairs; memory is worth 21 points on an eight-pair board while every other skill together is worth about one; and merit can take a category title from a higher score, so a record score can go down by design — which happens in 26 per cent of standing record pairs.
The sequences
| Sequence | What it counts |
|---|---|
| A389262 | The Collective Survival Threshold, contributed by the author: the least number of choices that keeps the hundred-prisoners loop strategy above one half. Its first singletons fall on the Fibonacci primes two, three and five. |
| A000041 | The partition numbers, which count the shapes a shuffle can take. |
Related work
The loop strategy behind Part II is the same structure that appears in Star Face as a spin-off, and the author's other papers — Coefficial Divisibility, The VIO Formula and The English Cube and Metatron Numbers — share its concern with what symmetry and counting can settle.
Download the collected volume (PDF, 51 pages)
Or simply play it grego.hu/limbo