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.