Talk:Coolout Conjecture

The phrase "internally consistent with being part of a still life" is rather vague. I think it should be rephrased to make it clearer which patterns are or are not eligible. 77topaz (talk) 03:58, 31 January 2018 (UTC)

There may not be sufficient documentation of the conjecture to determine this precisely. However, I believe the natural reading would be "every 3×3 area of the partial (or smaller, if overlapping with the partial's edges) is capable of occurring in a still life".

Schroeppel's original counterexample, and its 5×2 version, suffice to disprove actually even a stronger condition of 4×4-consistency (their 4×2 left ends could be extended into a shillelagh and hook with tail respectively) or also e.g. range 2 von Neumann neighborhood consistency. --Tropylium (talk) 11:35, 15 July 2026 (UTC)

5×5 consistency seems to not have been disproven yet.

If a partial is allowed to have indentations in it, we could craft examples of the sort shown on the left here, where the thickness-2 white/blue area with a П shape is the partial (green cells are casing). 5×5 overlaps with it can be stabilized (as shown on the right), but the whole thing cannot, since this would require both of the deep corner cells shown in yellow to be present.

x = 28, y = 6, rule = LifeHistory
.A.A2.2A5.A.A2.A2.2A.A2.A$AB5CBA3.AB4CA2.A.5C$A7BA3.A5B5.5B$.CBE.EBC
5.CBEA6.B.A.B$ACBA.ABCA3.ACB2A6.BA.AB$4.A20.A!

However this strikes me as "unsportsmanlike", and probably a partial for the purposes of (some generalized version of) this conjecture should be at least convex. --Tropylium (talk) 12:50, 15 July 2026 (UTC)

… disregard the example, I am failing basic counting today and the 2001 counterexample does disprove 5×5-consistency; 6×6 would be the current target (or perhaps 6×2; presumably if m×m has a counterexample, then already m×2 does). --Tropylium (talk) 13:12, 15 July 2026 (UTC)

Anyway, having now had my coffee and following more closely on the tails of the original construction, here are a 8×2 partial that disproves 7×n consistency, a 11×2 partial that disproves 10×n consistency, and a 14×2 partial that disproves 13×n consistency:

x = 31, y = 27, rule = LifeHistory
16.A$15.A.A$15.A2.A$2CB2CB2C4.2CB2CBC$CBC2BCBC4.CBC2BCB$3.2A10.2A5$
19.2A$18.A2.A2.2A$18.A2.A2.A$2CB2CB2CB2C4.2CB2CB2CBC$CBC2BC2BCBC4.CBC
2BC2BCB$3.A.A.A10.A.A.A$4.A.A12.A.A$5.A14.A3$22.2A4.A$21.A2.A2.A.A$
21.A2.A2.A2.A$2CB2CB2CB2CB2C4.2CB2CB2CB2CBC$CBC2BC2BC2BCBC4.CBC2BC2BC
2BCB$3.A.A2.A.A10.A.A2.A.A$4.A4.A12.A4.A!

This construction will generalize to arbitrarily long versions, thus showing that no finite covering set of stabilizable regions is sufficient for the overall stabilizability of a convex still life partial. --Tropylium (talk) 13:58, 15 July 2026 (UTC)