The 6-Square Pieces: with Subtractions

Here we gather constructions with the 6-pieces that use a subset of the 35.

W. E. Lester (FCR Oct/Dec 1938 ¶3399, ¶3400 and Dec/Feb 1938/9 ¶3478, ¶3479) showed how the 20 asymmetric 6-pieces could be used to form rectangles 5×24, 10×12, 6×20 and 8×15.

The second 10×12 and 8×15 examples are 'degenerate hollow rectangles' by T. R. Dawson. The 10×12 being of width 5, with 2-unit cut (FCR Jun/Aug 1948 ¶7726). The 8×15 being of width 4, with central 7-unit cut (FCR Feb/Apr 1948 ¶7593).

Dawson also showed (¶3401 and ¶3480) how these pieces can form two simultaneous rectangles, 5×12 or 6×10 (within the dotted outlines) that can be combined to solve the 5×24, 6×20 and 10×12 cases, with a bisecting fault line.

T. R. Dawson (FCR Feb/Apr 1948 ¶7593). The asymmetrical [6]s in hollow rectangles, path width = 4, sizes 11×12, 10×13, 9×14. (For degenerate 8×15 case see above).

T. R. Dawson (FCR Jun/Aug 1948 ¶7726). The asymmetrical [6]s in hollow square 11×11, path width = 5, with unit hole. (For degenerate 10×12 case see above).

F. Hansson (FCR Apr/Jun 1946 ¶6762). The 24 evenly-chequered pieces can be used to form a rectangle 6×24, with no crossroads.

F. Hansson (FCR Apr/Jun 1946 ¶6765). The 24 evenly-chequered pieces can also form a square 12×12, with no crossroads. Hansson, in the solution to HDB's ¶7000, also provided another 12×12 solution with one crossroad at the centre (it is described as a limiting case of HBD's hollow rectangles, but this is not correct). The third example, shown by W. Stead (FCR Dec 1954 ¶33), is a 12×12 with inset 6×6.

F. Hansson (FCR Apr/Jun 1946) showed how the 24 evenly-chequered pieces can also be used to form sextuplicated copies of the other 4-square shapes, with no crossroads and no cuts at reentrant angles.

F. Hansson (FCR Apr/Jun 1946 ¶6763).

F. Hansson (FCR Apr/Jun 1946 ¶6764).

F. Hansson (FCR Apr/Jun 1946 ¶6766).

H. D. Benjamin (FCR Oct/Dec 1946 ¶7000) The evenly chequered [6]s in hollow rectangles of width 4 units: 13×13 with 5×5 hole, 12×14 with 4×6 hole, 11×15 with 3×7 hole. (Other cases done by HDB but not published were 10×16, 9×17 and 8×18.

H. D. Benjamin (FCR Aug/Feb 1950/1 ¶8795-6) half-hollow rectangles of width 4 units: 10×17 and 11×14.

The 24 in Simultaneous Shapes

H. D. Benjamin (FCR Dec/Feb 1948/9 ¶7939). The evenly chequered [6]s in paired L-shapes which will form hollow or half hollow rectangles. Width = 4, sizes 5×17 and 10×12.

H. D. Benjamin (FCR Jun/Aug 1949 ¶8144). Width = 6, sizes 7×11 and 8×10. Two other cases were not shown.

The above four cases were presented in FCR as 'bisected half-hollow rectangles'. Here I have preferred to separate the two parts, to emphasise that they can be put together in other ways.

F. Hansson (FCR Apr/Aug 1955 ¶10,201). The 24 even-chequered pieces, plus the odd-chequered cross-shaped 55(6,32)-piece, in a fivefold copy of the same piece.

G. J. Boucher (FCR Oct/Feb 1955/6 ¶10,347). The 24 even-chequerd pieces, plus the odd-chequered 51(6,22)-piece, in a fivefold copy of the same piece.

G. J. Boucher (FCR Feb/Jun 1956 ¶10,439). The 24 even-chequered pieces, plus the odd-chequered 27(6,7)-piece, in a fivefold copy of the same piece.

F. Hansson (FCR Aug/Dec 1955 ¶10,303). The 24 even-chequerd pieces, plus a duplicate of the 56(6,33)-piece, forming a fivefold copy of the same piece.

These patterns use all the 6-pieces that cannot be cut from a 3×3 square. There are 27 such pieces, 19 evenly chequered and 8 oddly chequered, so can be arranged in rectangles.

H. D. Benjamin (FCR Dec/Jun 1939/40 ¶4143) in a rectangle 6×27.

H. D. Benjamin (FCR Feb/Apr 1938 ¶3089) a two-square rectangle 9×18 without crossroads. "A beautiful dissection by HDB which he later embodied in a magnificent rug" (FCR December 1954).

W. Stead (FCR Feb/Jun 1956 ¶10451-2). Rectangles formed by (a) the '13 line-pieces', 6×13, (b) the 14 'tree pieces', 6×14,

W. Stead (FCR Feb/Jun 1956 ¶10451-2) - (c) both sets combined, 9×18, domino.

The other 8 pieces Stead called 'clumps'.

The following 11 diagrams are the solutions by G. Fuhlendorf to a problem by F. Hansson (FCR April/June 1937 ¶2698) to cut a 12×17 rectangle into 34 [6]s, omitting one.

There are 11 of the [6]s which may be omitted, the 'odd'-chequered ones.