The best-known puzzles with the 12 differently shaped 5-square pieces are based on the fact that the smallest square within which the 12 pentominoes can be fitted is the 8×8 chessboard, leaving four squares uncovered. We consider possible arrangements of the four squares.
The results reported below were all found by pre-computer methods, however it is now possible to solve many cases by using computer programmes such as that provided at http://math.hws.edu/xJava/Pentominos_old/ which constructs a dissection when any four vacant squares are specified.
First we consider arrangements with the uncovered squares arranged to form a square. In the first four they form indentations in the edge so that the pentominoes form a connected shape without holes. The first is by W. E. Lester and B. Zastrow, independently in solving (PFCS Aug/Oct 1935 ¶1923). This is in two equal halves. The other three are from Pentominoes (1981) by S. Farhi.
The next 5 show the other centro-symmetric cases. The first two are by P. B. van Dalfsen and W. Stead, independently (FCR Jun/Oct 1953 ¶9602(a), ¶9603). The third is by W. Stead (FCR Dec 1954 ¶16). The fourth and fifth are from Pentominoes (1981) by S. Farhi. Finally an off-centre example from W. Stead's manuscripts, and a centred 'diamond' by W. Stead (FCR Dec 1954 ¶15).
I suggested in Chessics #28 that readers might like to try other non-centrosymmetric placements of the four squares, and asked: How many cases are there?
The next diagrams show four squares at the corners of a 3×3. The first is due to D. H. Hersom (FCR Apr/Jun 1938 ¶3163). The others all use the cross-shape to fill the rest of the 3×3. The second was proposed by W. Stead (FCR Oct 1956 ¶10637), but due to discontinuation of the column, no solution was published, however the solution appears in his manuscript books. The next two are my own work (Chessics #28, 1986). The solution centred at c6 is due to F. Hansson and D. H. Hersom (independently, in FCR Jun/Aug 1939 ¶3743). The last two are more of my own work, G. P. Jelliss (Chessics #28, 1986).
Here we consider cases where the squares are separate but may be diagonally adjacent. The first of the 8 diagrams below, showing the four spaces in a diagonal line, is by D. H. Hersom (FCR Dec/Feb 1937/8 ¶3031). The next three are my own work (Chessics #28, 1986). The first and last in the next row are by W. Stead (FCR Dec 1954 ¶13-14). The other two are my own work, G. P. Jelliss (Chessics #28, 1986).
W. Stead (FCR Jun/Oct 1956 ¶10,544-6). The voids mark squares which when occupied by Rook + Nightrider combinations guard all 64 squares. The first case strictly belongs in the previous section.
I sent six of these 'mask' problems to the Journal of Recreational Mathematics on 20th April 2003, but have heard no more of them. There are eighteen possible cases (on the assumptions that one cell is omitted in each quarter of the board, eye and mouth holes are separate, none abut an edge, and they form an axially symmetric pattern.
Nine are left for readers to try to solve. I show these dissections in a slightly different form, using a heavy line to indicate the external and internal 'edges' of the dissection, since the four missing squares are imagined to be holes cut through the square mask to allow eyes to see and mouth to speak through them.
The first dissection on this page, by Lester & Zastrow, omitting the four corners, also divides the board into two equal parts. These can be rearranged to give two dominoes removed from opposite edges. The other four examples here are my own work (G.P.Jelliss July 2006).
This is a disregarded combination. Here are four of my own construction (July 2006).
Unsolved problem: Is it possible to dissect the board into two triangles, of 28 and 36 squares, with serrated cut running from a7/a8 to g1/h1? The 3-piece being in the smaller triangle and the 1-piece in the larger, preferably both internal to the triangles, or both in the right-angled corners, as in the first example above.