The Longer Way Home

In Bulgarian solitaire you take one card from every pile and gather them into a new pile, over and over, and the hand always settles into a loop, falling, when the cards are triangular, to a single staircase. Now take two cards from every pile. Or s. The rule barely changed, but the staircase gets steeper, and the slowest hand's climb home, which for one card takes at most k²−k moves, jumps to exactly k² the instant you take a second card, and stays there for every s after.

I · Take more than one card

Move the s slider to change how many cards you take from every pile each move. The hand below is the slowest possible start for that many cards and that value of s. Press play and watch it climb to its home; the faint outline is the staircase it's reaching for, and it gets steeper as s grows.

Once the hand turns gold it has entered its cycle: from there it only ever repeats, so the number of moves up to that instant is its settling time. The rule, exactly: from every pile take min(height, s) cards (a short pile just gives up all it has), pile all the cards you took into one new pile, and re-sort. For s=1 that is the original game.

II · The steeper staircase, and the longer climb

When the cards number s·T_k (an s-fold triangular number), every hand converges to one staircase, (sk, s(k−1), …, 2s, s): the ordinary staircase stretched by s. This chart plots, for the current s, the worst-case settling time at every number of cards. The bright dots are those s·T_k convergence points; they sit on the theorem curve.

Here is the surprise. For the original game (s=1) Igusa (1985) and Étienne (1991) proved the slowest hand of T_k cards reaches the staircase in exactly k²−k moves. But compute the same worst case one card heavier (s=2, cards 2·T_k) and it is k². Not k²−k, not k²+k: exactly k². And it stays k² for s=3, 4, 5, and up: every heavier game shares the same worst-case climb, and only the original sits k moves below it.

III · What is charted, and what isn't

This game is not new, and most of its structure is already proved. Henrik Eriksson posed the take-s variant in 1981; Brian Hopkins gave it a full modern treatment (INTEGERS 24A, 2024), where our move is his operation H_s. He proves the recurrent hands are counted by a generalized binomial coefficient: for s=2 that is the trinomial triangle, so the count of recurrent hands at a square number of cards is the central trinomial coefficient 1, 3, 7, 19, 51, 141, …. He gives closed forms for the number of loops and the Garden-of-Eden hands too. The engine here reproduces all of those exactly before it claims anything, so you can trust it.

What Hopkins's paper does not settle, and what a search of the literature and OEIS turns up empty, is the settling time: how many moves the game takes to get to its home. The recurrent set is charted; the road in is not. So that is what this page computes and stages:

• The k² law. Worst-case settling at s·T_k is k² for every s≥2 and k²−k for s=1: a clean generalization of Igusa's bound with a +k jump at the first step. Verified for s=1…7, k up to the reach of exact enumeration; offered as a checked conjecture, not yet a proof.
• The settling sequences. The total settling time (summed over all p(n) hands) and the worst-case settling time, for s=2 and s=3, are each absent from the encyclopaedia. They are staged for a citable deposit.

The honest line is the whole point: the recurrent-set facts are Hopkins's theorems, reproduced here, not claimed; the settling results are computed exactly, verified every way the engine can be turned, and marked conjecture where they are not yet proved.