4-Square Pieces: Tetrominoes

B. Larsson (Fairy Chess Review April 1937, solution to ¶2623) states that there are 117 ways of dissecting a 4×4 board into 4 pieces each of 4 squares (includng rotations and reflections). This is correct. He also says that there are 21 fundamental forms, however this was probably a misprint as I make it 22, as reported in Chessics #28 and shown below. One case is unaltered by rotation or reflection, four cases can occur in 2 orientations, seven cases in 4 orientations, and ten asymmetric cases in 8 orientations. A dissection into 4 different 4-square pieces is impossible.

G. P. Jelliss (Chessics #3, 1977). There is just one way in which the chessboard can be divided into 16 square pieces 2×2, namely in the manner of Grid Chess (which was of course invented by Walter Stead, the dissection expert). It is impossible to dissect the board into 16 of the S-shaped tetrominoes, but the other three allow such dissections in more than one way. I considered the question of placing the minimum number of pieces of a given shape so that the positions of the other pieces, all of the given shape, were fixed. For the I-piece two are sufficient (e.g. those covering b5 and g5 in the solution below). For the T-piece four are needed (e.g. those at b2, b7, g2,g7 in the solution below). For the L-pieces, which present the most difficult case, four are also sufficient (e.g. those at a7, b1, g8, e3 in the solution below).

G. P. Jelliss (Chessics #3, 1977). Another problem I posed is: Find a dissection of the chessboard into 16 tetrominoes all of the same shape so that there are no 'crossroads' and no rectangles other than the complete square. I found 12 distinct solutions. The first three are symmetric—the second and third differing only by rotation of the central pair.

In a 'dissection' problem we usually imply that all the pieces are to be different. This is as opposed to a 'tessellation' problem, where any number of pieces of the same shape can be used.

The five 4-pieces cannot form a rectangle 4×5, or 2×10, because if the pieces are imagined to be placed on a chequered board the T-piece will cover 3 squares of one colour and 1 of the other colour, whereas all the other pieces will cover 2 of each colour. Any pattern covered must have a colour difference of 2 squares of one colour more than the other colour. On the 4×6 rectangle we can place the five 4-pieces and leave four single squares or a T-piece space.

H. D. Benjamin (FCR Dec/Feb 1937/8 ¶3026-30, June/August 1938 ¶3228) investigated arrangements of the five [4]s on a 5×5, leaving a single [5] space. There is one solution with an internal pentomino. The next four examples show the T-tromino enclosed or touching one edge.The last two have all [4]s and the [5] abutting the edges.

H. D. Benjamin (FCR Aug/Oct 1940 ¶4567) puts the [4]s on a 5×6 board to leave two [5]s of the same shape in symmetric formation. The T-piece touches 0, 1, 2 edges respectively.

W. E. Lester (FCR Apr/Jun 1948 ¶7652). Form a 36-cell triangle with two sets of [4]s, omitting one. The fifth case is easily proved impossible by a chequer count.

As we noted in the introduction, it is not obligatory to use only pieces with a given number of squares. Many interesting results can be shown by using specially selected pieces of a given number of squares, or by combining sets of pieces of different numbers of squares. At the end of each section on dissections using n-square pieces we will include a selection of examples using them in combination with those of smaller size.

The pieces of 1, 2 and 3 squares will form a 3×3 square.

Dorothy R. Dawson (daughter of T. R.) showed (PFCS Feb/Apr 1936 ¶2173-6) that the pieces of 2, 3 and 4 squares will cover a rectangle 2×14 (this example has the 2×2 as near to the centre as possible).

She also showed the 4×7 case. The first 4×7 has all on edge. This was one of about 50 results obtained by Dorothy. The next two show two 4-square pieces not on edge.

T. R. Dawson (FCR Aug/Oct 1946 ¶6883) gave the fourth 4×7 above, describing it as a limiting case of a 'hollow rectangle' (the 3-unit cut in the centre being regarded as a 'degenerate hole'). The and 5×6 with 1×2 hole, below, is the non-degenerate case.

T. R. Dawson (FCR Feb/Jun 1951 ¶8971). Maximum area (29 squares) enclosed by the 1 to 4 pieces.