1. The area bound
Pieces with disjoint interiors inside a square of side L cover at most its area, whatever their rotations. A triangle has area 1/2 and an L-tromino area 3, so
P89: n/2 ≤ L², P91: 3n ≤ L².
Equality forces the pieces to tile the square with no gap, so the bound is the optimum exactly when such a tiling exists.
2. P89: n = 4, 8, 16, 18
n = 4: cut the square of side √2 along both diagonals. n = 8: cut each cell of the 2 × 2 square along a diagonal. n = 16: four copies of the n = 4 square form the square of side 2√2. n = 18: cut each of the nine cells of the 3 × 3 square. The optima are √2, 2, 2√2 and 3.
3. P91: n = 12, 27
n = 12: the bound is 6, attained by six 2 × 3 rectangles, each made of two L-trominoes. n = 27: the bound is 9; the 9 × 9 tiling in Friedman’s table is the site’s reference certificate, verified exactly.
4. What stays open
Elsewhere the bound is not attained, and a gap in one arrangement does not rule out a smaller square for another. P91 n = 3 has the same bound 3, but the 3 × 3 square cannot be tiled by L-trominoes. P89 n = 1, 2 and P91 n = 1 were closed on 22 September by the diameter argument.
Credit and precision
hadsajnc #146 pointed out the full tilings for P91 on 25 September; this adopted contribution earns one permanent +2 proof award. NUE_13 #31 made the same observation for P89 the same day. It is the area argument of their P90 proof adopted on 18 September, so it is credited without a second award. 😰 #106 later wrote out the inequality and is credited, without a separate award. The constructions are in Friedman’s tables; no historical priority is claimed.
The six instances close through the existing proven-optimal mechanism, with history retained. √2 and 2√2 are irrational, and the normalized grid certificates of the others carry rounding, so no stored decimal record is certified grid-optimal.
Original submission (hadsajnc) ↗ · NUE_13 ↗ · 😰 ↗ · Friedman (tans) · Friedman (L's) · News