P89 n = 4, 8, 16, 18 and P91 n = 12, 27: full tilings
The pieces’ total area bounds the square from below, and these six rows have tilings with no gap, so the bound is the optimum. hadsajnc posted it for P91 (+2); NUE_13 made the same observation for P89, the area argument already awarded for P90, and 😰 wrote out the inequality; both are credited.
P90 n = 1: one domino
However the domino is turned, one of its two axis projections is at least 2. LittleChasa’s argument, with a concavity step filled in (+2).
P82 m = 6, n = 3: three circles in a hexagon
A centroid identity shows that three centres in the inset hexagon have a pair too close unless r ≤ √3/4. NUE_13’s proof, generated by DeepSeek as they disclosed (+2).
P17 n = 6: six points in a rectangle
Two strips hold three points each, and a band width that must shrink below itself gives the contradiction: the optimum is 1, attained exactly by the 2 × 3 grid. NUE_13’s emailed proof, generated by DeepSeek as they disclosed (+2).
Awards
Four new contributions each earn one permanent +2 proof award: NUE_13 two (P82, P17), hadsajnc one (P91), LittleChasa one (P90). One argument earns one award per contributor, so the repeated P89 area argument and the restated inequality are credited without a second award. No verifier, score unit or historical submission changes.