Sources page biographical material

5.A.3. ROLLING PIECE PUZZLES
Here one has a set of solid pieces in a tray and one tilts or rolls a piece into the blank space.
Thomas Henry Ward.

UK Patent 2,870 -- Apparatus for Playing Puzzle or Educational Games. Provisional: 8 Jun 1883; Complete as: An Improved Apparatus to be Employed in Playing Puzzle or Educational Games, 6 Dec 1883. 3pp + 1p diagrams.

US Patent 287,352 -- Game Apparatus. Applied: 13 Sep 1883; patented: 23 Oct 1883. 1p + 1p diagrams. Hexagonal board of 19 triangles with 18 tetrahedra to tilt.

George Mitchell & George Springfield. UK Patent 6867 -- A novel puzzle, and improvements in the construction of apparatus therefor. Applied: 16 Mar 1897; accepted: 5 Jun 1897. 2pp + 1p diagrams. Rolling cubes puzzle, where the cube faces are hollowed and fit onto domes in the tray. Basic form has four cubes in a row with two extra spaces above the middle cubes, but other forms are shown.

Sven Bergling invented the rolling ball labyrinth puzzle/game and they began to be produced in 1946. [Kenneth Wells; Wooden Puzzles and Games; David & Charles, Newton Abbot, 1983, p. 114.]

Ronald Sprague. Unterhaltsame Mathematik. Vieweg, Braunschweig, 1961. Translated by T. H. O'Beirne as: Recreations in Mathematics, Blackie, London, 1963. Problem 3: Schwere Kiste, pp. 3-4 & 22-23 (= Heavy boxes, pp. 4-5 & 25-26). Three problems with 5 boxes some of which are so heavy that one has to tilt or roll them.

Gardner. SA (Dec 1963). = Sixth Book, chap. 8. Gives Sprague's first problem.

Gardner. SA (Nov 1965). c= Carnival, chap. 9. Prob. 1: The red-faced cube. Two problems of John Harris involving one cube with one red face rolling on a chessboard. Gardner says that the field is new and that only Harris has made any investigations of the problem. The book chapter cites Harris's 1974 article, below, and a 1971 board game called Relate with each player having four coloured cubes on a 4 x 4 board.

Charles W. Trigg. Tetrahedron rolled onto a plane. JRM 3:2 (Apr 1970) 82-87. A tetrahedron rolled on the plane forms the triangular lattice with each cell corresponding to a face of the tetrahedron. He also considers rolling on a mirror image tetrahedron and rolling octahedra.

John Harris. Single vacancy rolling cube problems. JRM 7:3 (1974) 220-224. This seems to be the first appearance of the problem with one vacant space. He considers cubes rolling on a chessboard. Any even permutation of the pieces with the blank left in place is easily obtained. From the simple observation that each roll is an odd permutation of the pieces and an odd rotation of the faces of a cube, he shows that the parity of the rotation of a cube is the same as the parity of the number of spaces it has moved. He shows that any such rotation can be achieved on a 2 x 3 board. Rotating one cube 120^o about a diagonal takes 32 moves. If the blank is allowed to move, the the parity of the permutation of the pieces is the parity of the number of spaces the blank moves, but each cube still has to have the parity of its rotation the same as the parity of the number of spaces it has moved. If the identical pieces are treated as indistinguishable, the parity of the permutation is only shown by the location of the blank space. He suggests the use of ridges on the board so that the cube will roll automatically -- this was later used in commercial versions. He gives a number of problems with different colourings of the cubes.

Gardner. SA (Mar 1975). = Time Travel, chap. 9. Prob. 8: Rolling cubes. This is the first of Harris's problems. Computer analysis has found that it can be done in fewer moves than Harris had. Gardner also reports on the last of Harris's problems, which has also been resolved by computer.

A 3 x 3 array with 8 coloured cubes was available from Taiwan in the early 1980s. It was called Color Cube Mental Game -- I called it 'Rolling Cubes'. The cubes had thick faces, producing grooved edges which fit into ridges in the bottom of the plastic frame, causing automatic rolling quite nicely. I wonder if this was inspired by Harris's article.

John Ewing & Czes Kośniowski. Puzzle it Out -- Cubes, Groups and Puzzles. CUP, 1982. The 8 Cubes Puzzle, pp. 58-59. Analysis of the Rolling Cubes puzzle. The authors show how to rotate a single cube about a diagonal in 36 moves.