History and Rules
This game with very simple rules but complicated play is the distinctive board-game of the Maoris of New Zealand. The game is described in R.C.Bell's Board and Table Games, volume 2 (Oxford University Press 1969), but I first encountered it in the Steve Nichols magazine Games Monthly (November 1988, pages 32-33) where the rules, some history and a large diagram of the board were given but no analysis. The board as shown there is in the form of an eight-pointed star, with no connecting lines round the circumference, and oriented with the radial lines horizontal, vertical and at 45 degrees. However the symmetry of the opening position and the fact that moves round the circumference are permitted means that the diagram as shown below makes the rules clearer.
The game is mentioned at the end of the history section in the book What is Mathematics, Really? (by Reuben Hersh 1998) where, on p.231, a complex figure showing 'The flow of the game of mu torere' is quoted, without much explanation, from another book, Ethnomathematics by Marcia Ascher (Brooks/Cole, San Francisco 1991). This shows 92 numbered circles connected by curved and polygonal lines to form a graph with 180° rotary symmetry. This stimulated me to revisit my earlier notes and complete the analysis that follows.
The 46 Positions
The number of geometrically distinct positions possible (disregarding rotations and reflections) is 46. Since each of these can have black or white to move we arrive at the total 92 in the above-mentioned diagram. The 46 are made up of 8 with centre vacant, 19 with a black piece in the centre and 19 with white there. If we also group together positions that are complements of each other, i.e. with black and white interchanged, we arrive at 26 basic positions, which are conveniently lettered A to Z.
The above 27 diagrams show all the positions with centre vacant or occupied by black. The complementary positions we denote by the same letters with * attached to indicate interchange of colours. Of the 8 positions with centre vacant, two, C and C*, are complements of each other, while the other six, A, B, D, E, F, G, are self-complementary (A* = A and so on). If we take each letter as indicating the position with black to move (so that A is the opening position) then we can denote the same position but with white to move by the prefix ~. We now have a notation for all possible positions. The stalemate positions to aim for are ~X, X*, ~Y, Y*, ~Z, Z*.
A Superstalemate Puzzle in Mu Torere
The shortest game ending in stalemate follows the unique sequence A, ~H, I, ~B, I*, ~H*, Z* (3 pairs of moves) where on the last move white stalemates black. This is a 'superstalemate' in that all the black pieces are in one group. My puzzle is: Under the restriction that a player who can play a stalemating move must do so (i.e. 'reflex' play), find the unique sequence leading to black superstalemate of white.
The solution is given at the end of this article.
Note that the six positions X, Y, Z and ~X*, ~Y*, ~Z* are in fact impossible to reach in play, since they have no move leading into them. If we move the middle piece in Z, Y, X out to the rim we get positions A, B, C respectively, but the reversals of these moves are illegal. The Z-Z loop move can never be made since it is not black's move and white is stalemated so the game has ended (and similarly for the complementary move Z*-Z*). All possible transitions by black moves are listed below (the ~s are omitted). To find white move transitions, operate with * on all letters, bearing in mind that ** cancels out and that A, B, D, E, F, G are unaffected. (For example N - C*, O becomes N - C, O* while N* - K* becomes N - K.)
Table of transitions by black moves:
|A - H||H - A, Z||H* - I*||R - T, E||R* - W*|
|B - I, J, K||I - B||I* - H*, L*||S - E||S* - O*, P*|
|C - L, M||J - B, Y||J* - P*||T - E, R||T* - Q*|
|C* - N, O||K - B, Y||K* - N*||U - E||U* - M*, V*|
|D - P, Q||L - C||L* - I*||V - F, V||V* - U*|
|E - R, S, T, U||M - C, X||M* - U*||W - G||W* - R*|
|F - V||N - C*, O||N* - K*||X - C, M||X* - 0|
|G - W||O - C*, N||O* - S*||Y - B, J, K||Y* - 0|
|-||P - D||P* - J*, S*||Z - A, H||Z* - 0|
|-||Q - Q, D||Q* - T*||-||-|
Beginning at A black's only two moves both lead to versions of ~H and white's only reply gives I. Then black's only move gives ~B and white now has three choices, leading to I*, J*, K*. At this point, on his third move, black has his first chance to make an error; moving from I* to ~H* allows white to reach Z* and win (or alternatively white can make another error, taking the position back to A). On the other hand black could win by playing (3) I* to ~L* then white must go to C* and black can play (4) ~N (not ~O) and white must reply K and black has (5) ~Y stalemating white. This analysis indicates that white must avoid I* on his second move and play J* or K*.
At this point in the original article in issue 17 I gave a chart of the possible moves in games, 9-deep, but it was not very clear, and is superseded by the simpler chart below which appeared in issue 18.
After publication of the above, Ken Whyld sent details of articles in the US Mathematics Magazine by Marcia Ascher (vol.60 1987 pp.90-100), and by Philip D. Straffin Jr (vol.68 1995 pp.382-6, vol.69, 1996 p.65) which also analyse Mu Torere. Their results agree with mine except that the rule that "a man between two of its own men cannot move to the centre" is applied only for each player's first two moves. This rule makes transitions A-Z, B-Y, C-X for black and A-Z*, B-Y*, C*-X* for white admissible. Since this weasely form of the rule ruins my nice superstalemate problem (the reflex rule forces C*-X*) I'm naturally against it.
The following is a compressed form of the network of moves (more elaborate versions are shown in the references cited above). Assuming that we start at A with black to move, the heavier lines represent black moves and the lighter lines white moves. However, upon entering any of the six self-complementary nodes A, B, D, E, F, G this convention can, if desired, be reversed (heavy lines then represent white moves and light lines black moves, all the node-names being converted to their complements, that is C becomes C*, O* becomes O, etc). This convention applies until a self-complementary node is reached again.
The dotted lines represent the optional moves, prohibited in my version. The circles at V and Q represent loop moves, in which the turn to play changes but the position is effectively the same. The network diagram indicates that F and G are obvious alternative opening positions.
Solution to the Superstalemate Puzzle
The sequence of positions leading to black's superstalemate of white under reflex conditions is: A, ~H, I, ~B, K*, ~N*, O*, ~S*, E, ~S, O, ~C*, L*, ~I*, B, ~I, H, ~Z as illustrated here:
The stalemate is reached on black's 9th move. There are routes via ~J* or ~K* instead of ~I* but these permit white to stalemate by playing to Y* instead of B. The problem is to get back to B with black to move instead of white, i.e. tempo loss, so we seek the shortest odd-length circuit.