Sources page biographical material

5.AG.2. HUNGARIAN RINGS, ETC.
New section. Much to be added.
William Churchill. US Patent 507,215 -- Puzzle. Applied: 28 May 1891; patented: 24 Oct 1893. 1p + 1p diagrams. Two rings of 22 balls, intersecting six spaces apart.

Hiester Azarus Bowers. US Patent 636,109 -- Puzzle. Filed: 16 Aug 1899; patented 31 Oct 1899. 2pp + 1p diagrams. 4 rotating discs which overlap in simple lenses.

Ivan Moscovich. US Patent 4,509,756 -- Puzzle with Elements Transferable Between Closed loop Paths. Filed: 18 Dec 1981; patented: 9 Apr 1985. Cover page + 3pp + 2pp diagrams. Two rings of 18 balls, each stretched to have two straight sections with semicircular ends. The rings cross in four places, at the ends of the straight sections, so adjacent crossing points are separated by two balls. I'm not sure this was ever produced. Mentions three circular rings version, but there each pair of rings only overlaps in two places so this is a direct generalization of the Hungarian Rings.

David Singmaster. Hungarian Rings groups. Bull. Inst. Math. Appl. 20:9/10 (Sep/Oct 1984) 137-139. [The results were stated in Cubic Circular 5 & 6 (Autumn & Winter 1982) 9 10.] An article by Philippe Paclet [Des anneaux et des groupes; Jeux et Stratégie 16 (Aug/Sep 1982) 30-32] claimed that all puzzles of two rings have groups either the symmetric or the alternating group on the number of balls. This article shows this is false and determines the group in all cases. If we have rings of size m, n and the intersections are distances a, b apart on the two rings. Then the group, G(m, n, a, b) is the symmetric group on m+n-2 if mn is even and is the alternating group if mn is odd; except that G(4, 4, 1, 1) is the exceptional group described in R. M. Wilson's 1974 paper: Graph puzzles, homotopy and the alternating group -- cited in Section 5.A under The Fifteen Puzzle -- and is also the group generated by two adjacent faces on the Rubik Cube acting on the six corners on those faces; and except that G(2a, 2b, a, b) keeps antipodal pairs at antipodes and hence is a subgroup of the wreath product Z₂ wr S_a+b 1, with three cases depending on the parities of a and b.

Bala Ravikumar. The Missing Link and the Top-Spin. Report TR94-228, Department of Computer Science and Statistics, University of Rhode Island, Jan 1994. Top-Spin has a cycle of 20 pieces and a small turntable which permits inverting a section of four pieces. After developing the group theory and doing the Fifteen Puzzle and the Missing Link, he shows the state space of Top-Spin is S₂₀.