ç Index to Chessboard Dissection Problems

# All the [5]-Pieces and [4]-Pieces Combined

## Rectangles

The 4 and 5 square pieces have a total area of 80 squares, so will combine to form rectangles 4×20 or 5×16 or 8×10.

H. D. Benjamin (PFCS Feb/Apr 1938 ¶2199-2201) 4×20 showing first all pieces on edge, then only the 1×5 piece internal, then the 1×4 and 1×5 pieces internal. His next example (FCR Oct/Dec 1937 ¶2924) splits into two parts (4×8 + 4×12), every piece on an edge (see also the section on Multiple Shapes below.) His last example (FCR Jun/Aug 1939 ¶3742) also in two parts (4×9 + 4×11) will further subdivide into parts, along the heavily marked lines, that will recombine to form differently split rectangles (4×7 + 4×13) and (4×6 + 4×14).

H. D. Benjamin (FCR Oct/Dec 1936 ¶2483-5) 5×16 first showing all pieces on edge, then only the 1×5 piece internal, then the 1×4 and 1×5 pieces internal.

H. D. Benjamin (FCR Oct/Dec 1941 ¶4957-8, Apr/Jun 1942 ¶5139-40, Jun/Aug 1942 ¶5210-11, Aug/Oct 1942 ¶5270-1, Oct/Dec 1942 ¶5338-9, Dec/Feb 1942-3 ¶5397-8). A series of twelve 8×10 rectangles showing each [5]-piece internal and all the other pieces on edge; all have no crossroads.

H. D. Benjamin (PFCS Feb/Apr 1936 ¶2171, FCR Feb/Apr 1941 ¶4711, Apr/Jun 1941 ¶4774, Jun/Aug 1941 ¶4843, Aug/Oct 1941 ¶4893). A series of five 8×10 rectangles showing each of the [4]-pieces internal with all other pieces on edge. Only the first has a crossroad.

H. D. Benjamin. (PFCS Feb/Apr 1936 ¶2172, FCR Aug/Oct 1943 ¶5656, Apr/Jun 1944 ¶5927-8). Four 8×10 rectangles with minimum number of pieces (8) on edge, the rest (9) being internal. All but the first have no crossroads.

H. D. Benjamin (FCR Apr/Jun 1947 ¶7250). Several of the above 8×10 rectangles can be seen as 'degenerate hollow rectangles', having a two-unit cut in the centre. The first example here, path width = 4, was constructed to show that feature. [See section 5b for a related example by T. R. Dawson having a 1×1 central hole.]

F. Hansson (FCR Dec/Apr 1954/5 ¶10,068) constructed two 8×10 rectangles with three crossroads in symmetrical formation.

## Other Shapes

### Biaxial Symmetry

H. D. Benjamin and W. H. Rawlings (FCR Dec/Feb 1936/7 ¶2556-7) The [5]s and [4]s forming Greek crosses, in the shape of 5 squares each 4×4.

G. Fuhlendorf (FCR Aug/Oct 1936 ¶2410) The [5]s and [4]s forming a 'diamond' or 'diagonal square', diameter 13, with a cross-shaped central hole.

H. D. Benjamin (FCR Feb/Apr 1946 ¶6693) The [5]s and [4]s forming a 10×10 square with holes.

### Axial Symmetry

H. D. Benjamin (FCR Feb 1942 ¶5050). The [5]s and [4]s in a V for victory.

T. R. Dawson (FCR Feb 1942 ¶5052). The [5]s and [4]s in a framed V for victory.

H. D. Benjamin (FCR Feb/Apr 1947 ¶7158). The [5]s and [4]s forming a Christmas tree.

### Asymmetry

H. D. Benjamin (FCR Aug/Oct 1938 ¶3306). The [5]s and [4]s in an L-shape, all pieces on edge, no crossroads.

## Multiple Shapes

H. D. Benjamin (FCR Oct/Dec 1937 ¶2924), as noted above, arranged the [5]s and [4]s to form two rectangles 4×8 and 4×12 which can be assembled to form 6 different 5-square shapes quadrupled (1, 2, 4, 5, 7, 9), as shown here.

H. D. Benjamin (FCR Feb/Apr 1937 ¶2620) also used the [4]s and [5]s, every piece on an edge, to make two shapes that will fit together to form 8 of the 12 shapes of 5 squares. The shapes can be formed by keeping the L-piece fixed and moving the domino round it to various positions, as shown here, giving the sequence 2, 8, 4, 10, 3, 5, 11, 6, 4, and back to 2. The P-shape (4) can be formed in two different ways.

H. D. Benjamin (FCR Oct/Dec 1948 ¶7864). The [5]s and [4]s are arranged here to form pairs of L-shapes. Width = 4, sizes 5×9 and 6×8. These can be combined to form hollow or half hollow rectangles. (These were originally presented as 'bisected half hollow rectangles').

## All the [5]s and Some [4]s

### Omitting One [4]

H. D. Benjamin (FCR Feb/Apr 1943 ¶5418) formed an 8×10 using all the [5]s and [4]s, the 2×2 being central with cuts dividing the rest of the board into four equal parts. This diagram can also be interpreted as an 8×10 board with central 2×2 hole and the rest made up of all the [5]s and four of the [4]s, one in each quarter. With this interpretation four other cases are possible, as he showed later (FCR Apr/Jun 1943 ¶5498).

### Omitting Two [4]s

T. R. Dawson (FCR Apr/Jun 1947 ¶7250, Jun/Aug 1947 ¶7324). The [5]s and three [4]s in hollow rectangles 9×9 with 3×3 hole, 8×10 with 2×4 hole. Ten choices of the three [4]s are possible. The two omitted [4]s are shown at the top of each column. Three cases were shown. To these are added examples (right-hand column) in which the three [4]s used are the nonrectangular ones (G. P. Jelliss, June 2006).

T. R. Dawson (FCR Oct/Dec 1947 ¶7459). The [5]s and three [4]s in degenerate hollow rectangles 6×12 with 0×6 'hole', path width = 3. Ten choices of the three [4]s are possible. The two omitted [4]s are shown at the top of each column. Three cases were shown. Again the case is added where the three [4]s used are the nonrectangular ones (G. P. Jelliss, June 2006).