The [5]-Pieces on the Chessboard: the other 4 cells connected

We now turn to the cases where the four squares are all laterally connected and thus form a 4-piece (tetromino). The 5-pieces can be combined with any tetromino to cover the 8×8 board.

In the pages of Fairy Chess Review it was the usual practice of the editor, T. R. Dawson, who himself contributed many constructions of this type, to publish solutions to the 12[5]s + 1[4] problem in groups showing, where possible, all five of the 4-pieces. It is helpful to bear in mind that often only one or two moves of the 5-pieces can change a solution for one 4-piece into a solution for another 4-piece, or into another position of the same 4-piece.

Dawson made a little study of such transfers as he called them. He found a five-fold transformation (FCR Jun/Aug 1944 ¶6011-5) as shown in the first row below. However later (FCR Feb/Apr 1946 ¶6695-8) he seems to have forgotten this result and claimed that he could only find four-fold solutions. In drawing his solutions for this article I found that one of his four-piece solutions also in fact admits the fifth (I) case as well, as in the second row here.

His other four-piece solutions are as here. The first row uses the same pair of pieces (2 and 4) as above. The others use a different pair (3 and 4):

His problem 6699 shows five consecutive transfers:

Theorem: Dissections using the 12[5]s and 1[4] can be constructed to show each of the five 4-pieces in all their possible positions on the board, except where they cut off a corner.

We proceed to prove this by giving examples covering all possible positions of the pieces. I call this 'Dawson's Theorem' since he showed how to prove most of the cases in an economical manner by incorporating several cases in a single diagram. I number the cases as shown below. (This treatment is based on my article in The Games and Puzzles Journal #8+9 1988/9 p124.)

By translation (i.e. movement without rotation or reflection) each 4-piece can be moved so that the bottom left cell of its containing rectangle can occupy any of the numbered positions. The three impossible cases are with the T, S or L piece in the corner numbered 1.

Other positions of the 4-piece on the board can be converted into these numbered positions by translation, rotation or reflection. For instance moving the S to the top right corner of the board gives a position equivalent to case (1), but moving the T to the top right corner gives a position equivalent to case (7). The difference is due to the different symmetries of the two pieces, the S being unaltered by rotation and the T by reflection.

The following three diagrams, by T. R. Dawson (Fairy Chess Review, April 1937 ¶2648-50) show that the four squares can be left in 2×2 formation anywhere on the board (10 geometrically distinct positions). The L-shaped [5]-piece bordering two sides of the 2×2 can be rotated within the 3×3 area it occupies to provide other positions for the 2×2.

The following six positions, consist of four given by T. R. Dawson (FCR Aug/Oct 1943 ¶5657-two, Oct/Dec 1943 ¶5727, Feb/Apr 1944 ¶5860) showing inset rectangles 2×7, 3×8 and 4×6, plus two solutions of my own (The Games and Puzzles Journal Vol.1, #8+9, p.124, 1988/9). By reflecting the rectangles alternative positions of the tetromino are shown. Position 8 is shown by a transfer of the I-piece to the position marked by circles.

A proof for this case was given by Dawson (FCR Feb/Apr 1943 ¶5448), showing in these seven diagrams all 20 possible positions of the T. They are shown by rotations of the inset 3×3 and by some transpositions, as indicated by the circles.

Dawson proved the L-shaped tetromino case (FCR Apr/Jun 1943 ¶5499), with the following eight diagrams, showing all 41 possible positions by rotations and reflections of the 3×3 and some transpositions. The X-marks are positions reached by two transpositions.

A full treatment of the S case does not appear in FCR, but examples cover 12 cases. The S does not lend itself to formations that can easily be transformed to show other cases. In the following diagrams the first 9 are from FCR (the first two include a 3×8 rectangle that can be rotated or reflected to give the second case, the third is from the 5-fold transition shown at the end of the introductory section above). The tenth diagram, showing cases 16 and 19, is my own work. For the remaining six cases I made use of the programme provided at http://math.hws.edu/xJava/Pentominos_old/ which constructs a dissection when any four vacant squares are specified.

The 8×8 board can of course be regarded as a 4-fold magnification of the square 4-square piece 2×2. Frans Hansson (FCR Aug/Oct 1949 ¶8224) considered the analogous problem of forming an area the same shape as each of the other four four-pieces, with the added 4-piece being of the same shape as the whole.

Here R. J. French (FCR Dec/Jun 1939/40 ¶4144-8) uses the 12[5]s + 1[4] to form 64-square 'pyramids', with the 4-piece as near the apex as possible.