THE FOLD / RESPAWN / THE PHOENIX / THE BULGARIAN SOLITAIRE
THE BULGARIAN SOLITAIRE
any pile grinds down to the staircase
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Bulgarian solitaire is a deceptively simple card game on a number. Deal n cards into piles of any sizes; each move, take one card from every pile and gather them into a single new pile. When n is a triangular number n = 1+2+…+k, this process, from any starting configuration, always settles into the same fixed staircase {k, k−1, …, 1} — a stable attractor it can never leave.
LIT verified live: for triangular n up to 36, hundreds of random starting partitions all converge to the staircase, which is a fixed point of the move (window.__bulgarian). FIG no framing; deterministic integer dynamics.
LIT verified live: for triangular n up to 36, hundreds of random starting partitions all converge to the staircase, which is a fixed point of the move (window.__bulgarian). FIG no framing; deterministic integer dynamics.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-phoenix — whatever heap you start from, the same shape rises from the ashes: the staircase reforms every time. Bulgarian solitaire is that reforming. AVAN (AI) built the instrument: the take-one-from-each-pile move, the convergence loop, and the checks that the staircase is reached and is a fixed point.
Credit as content: popularized by Martin Gardner (1983); analyzed by Jørgen Brandt and others. The weave: David names the-phoenix; I run the deterministic move from many random partitions of a triangular n and confirm they all fall into the same staircase — a self-organizing fixed point.
Credit as content: popularized by Martin Gardner (1983); analyzed by Jørgen Brandt and others. The weave: David names the-phoenix; I run the deterministic move from many random partitions of a triangular n and confirm they all fall into the same staircase — a self-organizing fixed point.
3 ONE DIMENSION
Each step: remove one card from every pile, and those removed cards form one new pile. For triangular n, every start funnels to the staircase {k, k−1, …, 1}.
4 TWO DIMENSIONS · INTERACTIVE
Watch a random start funnel to the staircase step by step; the convergence checked over many starts.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: one attractor swallowing every start.
AVAN’s addition (the inverse-companion): reach one fixed shape from every starting heap by a single blind rule — take one from each pile, drop them as a new pile — and the staircase self-assembles. The inverse of ‘design the target and build it’ is ‘apply one local move everywhere and let the attractor emerge.’ Magenta is the many possible starts; green is the single staircase they all become. Order that assembles itself.
LIT Genuine Bulgarian solitaire (popularized by Martin Gardner 1983; analyzed by Jørgen Brandt and others). Verified live: for triangular n = k(k+1)/2 with k=1..8, 200 random partitions each converge under the take-one-from-each-pile move to the staircase {k,…,1} (window.__bulgarian.allConverge), and the staircase is confirmed a fixed point — deterministic integer dynamics (worst observed ~54 steps).
FIG No framing: the take-one-from-each-pile move, the convergence loop, and the reached-and-fixed-point checks run in-browser and agree. The AVAN inverse is honest — reaching one fixed shape from every start by a single blind local rule genuinely self-assembles the staircase rather than being designed; magenta is the many possible starts, green the single attractor they all become. Order that assembles itself.
FIG No framing: the take-one-from-each-pile move, the convergence loop, and the reached-and-fixed-point checks run in-browser and agree. The AVAN inverse is honest — reaching one fixed shape from every start by a single blind local rule genuinely self-assembles the staircase rather than being designed; magenta is the many possible starts, green the single attractor they all become. Order that assembles itself.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN