Jagged or skewed squares, particularly one made of 61 small squares with the central square missing, are part of pentomino puzzle history. Tiling the 60 square version with 12 pentominos was proved impossible in the 1980's.

Here is a small jagged square box composed of 25 small squares and a second box, with the middle square missing, composed of 24 small squares.

The 25 square version can be tiled by 5 pentominos, (F, N, P, W, and X), and the 24 square version by 4 hexominos, (A, B, C, and D). Both are presented as puzzles; the hexomino one is trivial; the pentomino one a bit harder.

Many additional sets of pieces that tile the small jagged square boxes can be created with combinations of the 5 pentominos, the 4 hexominos and a single "T" tetromino. The F, N, P, and W pentomino's can be replaced in turn by the "T" tetromino to make four sets of pieces that tile the jagged square box with 24 small squares.

One can also take the 4 hexominos in turn, combine them with the "T" tetromino and see if the combination can replace two of the pentominos in the tiling of the jagged square box with 25 small squares. Twelve such sets of pieces can be found. Solutions to all of these puzzles are given below.

MakerBot

MakerBot Replicator+

Yes

No

0.2 mm

In the image below the 4 hexominos are labeled A,B,C, and D. The five pentominos are identified, as is common, by letters of the alphabet F,N,P,W, and X. The captions below the puzzles show the pieces contained and the number of arrangements for each set of pieces. The T tetromino is colored red.

Show_all.scad displays answers to all of these problems. OpenSCAD code that constructs the jagged square boxes is provided. A pdf "key" is included for any who wish to construct the puzzles.

Further work: One can pretend that the 25 square jagged box has only 24 squares with an arbitrary square left unoccupied. IF so the X pentomino can be replaced and the FNPW pentominos plus the "T" tetromino will tile the box, the missing square offset from the center. One can substitute each of the four additional tetrominos for the "T" one and seach in the same fashion. At least one of the five possible sets of pentominos and any tetromino, even the straight one, will tile the box in this fashion. The uncovered square is often at an edge.