The Longest Way Home

Deal a handful of cards into piles. Take one card from every pile and set those aside as one new pile. Do it again. Again. This is Bulgarian solitaire, and no matter how you start it always ends up cycling — and when the number of cards is triangular (1, 3, 6, 10, …) it falls all the way to a single staircase and stays. The question with a surprisingly deep answer: how long can it take to settle?

I · Watch it settle

Pick a number of cards. The hand below is the slowest possible start for that number — the arrangement that takes the most moves before it ever repeats. Press play and count the moves until it locks into its loop.

cards = 10

move 0 — settling…

The faint outline is the staircase — the home it's reaching for. Once the hand turns green it has entered its cycle: from there it only ever repeats, so the count of moves up to that instant is its settling time.

II · The worst case, for every hand

For each number of cards, the dot is the worst-case settling time — the most moves any starting hand of that size needs before it first lands on its loop. The gold curve is the theorem.

triangular hands sit on the gold curve k²−k; the rest hang below.

When the cards number T_k = k(k+1)/2 — a triangular number — Brandt (1982) proved every hand reaches the one staircase, and Igusa (1985) & Étienne (1991) proved the slowest hand gets there in exactly k² − k moves, never more. Those are the bright dots, sitting precisely on the curve. The dim dots between them — when the number of cards isn't triangular, so home is a small loop rather than a single staircase — have no known formula. They are what this page computes.

The check — live, in your browser

Recomputing the whole game from scratch…

III · What is and isn't new here

Two honest layers. The engine that drew everything above first has to earn trust, so it reproduces five already-catalogued facts about this game exactly — the count of recurrent hands is Pascal's triangle (Étienne), the number of distinct loops is A037306, the unreachable "Garden of Eden" hands are A123975, the longest loop is A183110, and the staircase bound k²−k lands on the nose. Only then does it report what's uncharted:

The total settling time — add up, over all p(n) possible hands, how many moves each needs to settle — is a sequence absent from the encyclopaedia: 0, 0, 3, 3, 8, 33, 26, 41, 86, 267, … It has since been deposited, citable by DOI (10.5281/zenodo.21943897), calibrated against the five known sequences so its novelty is the only thing being claimed.

A note on the worst-case curve itself: that sequence (0,0,2,2,3,6,4,5,7,12,…) is also absent from the encyclopaedia, but it isn't independent — we verified through 61 cards that it equals A188160 (the catalogued "maximum steps to reach a period," which bundles in one whole loop) minus the longest-loop length, plus one. That small identity says something real — the slowest hand always drains into a longest loop, even when shorter loops exist — so we offer it as a checked conjecture, not a discovery. The honest line between "verified to 61" and "proven" is the whole point.