Sources page biographical material

6.G.1. OTHER CUBE DISSECTIONS
See also 6.N, 6.U.2, 6.AY.1 and 6.BJ. The predecessors of these puzzles seem to be the binomial and trinomial cubes showing (a+b)³ and (a+b+c)³. I have an example of the latter from the late 19C. Here I will consider only cuts parallel to the cube faces -- cubes with cuts at angles to the faces are in 6.BJ. Most of the problems here involve several types of piece -- see 6.U.2 for packing with one kind of piece.
Catel. Kunst-Cabinet. 1790. Der algebraische Würfel, p. 6 & fig 50 on plate II. Shows a binomial cube: (a + b)³ = a³ + 3a²b + 3ab² + b³.

Bestelmeier. 1801. Item 309 is a binomial cube, as in Catel. "Ein zerschnittener Würfel, mit welchem die Entstehung eines Cubus, dessen Seiten in 2 ungleiche Theile a + b getheilet ist, gezeigt ist."

Hoffmann. 1893. Chap. III, no. 39: The diabolical cube, pp. 108 & 142 = Hoffmann-Hordern, pp. 108-109, with photos. 6: 0, 1, 1, 1, 1, 1, 1, i.e. six pieces of volumes 2, 3, 4, 5, 6, 7. Photos on p. 108 shows Cube Diabolique and its box, by Watilliaux, dated 1874-1895.

J. G. Mikusiński. French patent. ??NYS -- cited by Steinhaus.

H. Steinhaus. Mikusiński's Cube. Mathematical Snapshots. Not in Stechert, 1938, ed. OUP, NY: 1950: pp. 140 142 & 263; 1960, pp. 179 181 & 326; 1969 (1983): pp. 168-169 & 303.

John Conway. In an email of 7 Apr 2000, he says he developed the dissection of the 3 x 3 x 3 into 3 1 x 1 x 1 and 6 1 x 2 x 2 in c1960 and then adapted it to the 5 x 5 x 5 into 3  1 x 1 x 3, 1 2 x 2 x 2, 1 1 x 2 x 2 and 13 1 x 2 x 4 and the 5 x 5 x 5 into 3  1 x 1 x 3 and 29 1 x 2 x 2. He says his first publication of it was in Winning Ways, 1982 (cf below).

Jan Slothouber & William Graatsma. Cubics. Octopus Press, Deventer, Holland, 1970. ??NYS. 3 x 3 x 3 into 3 1 x 1 x 1 and 6 1 x 2 x 2. [Jan de Geus has sent a photocopy of some of this but it does not cover this topic.]

M. Gardner. SA (Sep 1972) c= Knotted, chap. 3. Discusses Hoffmann's Diabolical Cube and Mikusiński's cube. Says he has 8 solutions for the first and that there are just 2 for the second. The Addendum reports that Wade E. Philpott showed there are just 13 solutions of the Diabolical Cube. Conway has confirmed this. Gardner briefly describes the solutions. Gardner also shows the Lesk Cube, designed by Lesk Kokay (Mathematical Digest [New Zealand] 58 (1978) ??NYS), which has at least 3 solutions.

D. A. Klarner. Brick packing puzzles. JRM 6 (1973) 112 117. Discusses 3 x 3 x 3 into 3  1 x 1 x 1 and 6 1 x 2 x 2 attributed to Slothouber Graatsma; Conway's 5 x 5 x 5 into 3  1 x 1 x 3 and 29 1 x 2 x 2; Conway's 5 x 5 x 5 into 3 1 x 1 x 3, 1 2 x 2 x 2, 1  1 x 2 x 2 and 13 1 x 2 x 4. Because of the attribution to Slothouber & Graatsma and not knowing the date of Conway's work, I had generally attributed the 3 x 3 x 3 puzzle to them and Stewart Coffin followed this in his book. However, it now seems that it really is Conway's invention and I must apologize for misleading people.

Leisure Dynamics, the US distributor of Impuzzables, a series of 6 3 x 3 x 3 cube dissections identified by colours, writes that they were invented by Robert Beck, Custom Concepts Inc., Minneapolis. However, the Addendum to Gardner, above, says they were designed by Gerard D'Arcey.

Winning Ways. 1982. Vol. 2, pp. 736-737 & 801. Gives the 3 x 3 x 3 into 3 1 x 1 x 1 and 6 1 x 2 x 2 and the 5 x 5 x 5 into 3  1 x 1 x 3, 1 2 x 2 x 2, 1 1 x 2 x 2 and 13  1 x 2 x 4, which is called Blocks-in-a-Box. No mention of the other 5 x 5 x 5. Mentions Foregger & Mather, cf in 6.U.2.

Michael Keller. Polycube update. World Game Review 4 (Feb 1985) 13. Reports results of computer searches for solutions. Hoffmann's Diabolical Cube has 13; Mikusinski's Cube has 2; Soma Cube has 240; Impuzzables: White -- 1; Red -- 1; Green -- 16; Blue -- 8; Orange -- 30; Yellow -- 1142.

Michael Keller. Polyform update. World Game Review 7 (Oct 1987) 10 13. Says that Nob Yoshigahara has solved a problem posed by O'Beirne: How many ways can 9  L trominoes make a cube? Answer is 111. Gardner, Knotted, chap. 3, mentioned this. Says there are solutions with n L trominoes and 9 n straight trominoes for n  1 and there are 4 solutions for n = 0. Says the Lesk Cube has 4 solutions. Says Naef's Gemini Puzzle was designed by Toshiaki Betsumiya. It consists of the 10 ways to join two 1 x 2 x 2 blocks.

H. J. M. van Grol. Rik's Cube Kit -- Solid Block Puzzles. Analysis of all 3 x 3 x 3 unit solid block puzzles with non planar 4 unit and 5 unit shapes. Published by the author, The Hague, 1989, 16pp. There are 3 non planar tetracubes and 17 non planar pentacubes. A 3 x 3 x 3 cube will require the 3 non planar tetracubes and 3 of the non planar pentacubes -- assuming no repeated pieces. He finds 190 subsets which can form cubes, in 1 to 10 different ways.

Nob Yoshigahara. (Title in Japanese: (Puzzle in Wood)). H. Tokuda, Sowa Shuppan, Japan, 1987. Pp. 68-69 is a 3^3 designed by Nob -- 6: 01005.