The Game of Mu Torere

The New Zealand game of mū tōrere is illustrated above with a beautiful handmade wooden board. The game seems to have been developed by the Māori people in response to the European game of draughts (checkers). Play is quite different from draughts, however. The game starts as shown above, with Black to move first. Legal moves involve moving a piece to an adjacent empty space:

  • along the periphery (kewai), or
  • from the centre (pūtahi) to the periphery, or
  • from the periphery to the centre, provided the moved piece is adjacent to an opponent’s piece.

Game play continues forever until a draw is called (by mutual consent) or a player loses by being unable to move. Neither player can force a win, in general, so a loss is always the result of a mistake. For each player there is one “big trap” and four “small traps.” This is the “big trap” (Black wins in 5 moves):

  
The board on the left is the “big trap” for White – Black can force a win by moving as shown, which leaves only one move for White.

  
Again, Black moves as shown, which leaves only one move for White.

  
Now, when Black moves as shown, White cannot move, which means that White loses.

Here is one of the four “small traps” for White. The obvious move by Black results in White losing (but avoiding this does not require looking quite so far ahead as with the “big trap”):

Here (click to zoom) is the complete network of 86 game states for mū tōrere (40 board positions which can occur in both a “Black to move” and a “White to move” form, plus 6 other “lost” board positions). Light-coloured circles indicate White to move, and dark-coloured circles Black to move, with the start position in blue at the top right. Red and pink circles are a guaranteed win for Black, while green circles are a guaranteed win for White. Arrows indicate moves, with coloured arrows being forced moves. The diagram (produced in R) does not fully indicate the symmetry of the network. Many of the cycles are clearly visible, however:


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Mathematics in action: “Skin groups”

There are three main (though closely related) branches of mathematics – the study of number, the study of shape, and the study of relationships. An interesting ethnomathematical example of the latter is the “skin group” system of the Lardil people of Mornington Island in Australia.


Lardil children

Similar systems (often with two groups, called moieties) can be found around the world (I first discovered the concept as a child, in the classic young adult science fiction novel Citizen of the Galaxy). Among the Lardil, there are eight groups, each associated with particular totemic creatures or objects:

Skin Group Name Totem
1 Burulungi Lightning
2 Ngariboolungi Shooting star
3 Bungaringi Turtle
4 Yugumari Seagull
5 Gungulla Grey shark
6 Bulunyi Crane
7 Bulyarini Sea turtle
8 Gumerungi Rock

Membership of a “skin group” implies a complex set of tribal obligations and taboos, but the most significant is that only certain kinds of marriages are permitted. Members of group 1 must marry people from group 2 (and vice versa), and similarly for the pairs 3/4, 5/6, and 7/8. All other marriages are considered to be incestuous.

We can define a mathematical function, the spouse function σ, that maps each person’s “skin group” to the “skin group” that their spouse must have: σ(1) = 2, σ(2) = 1, σ(3) = 4, σ(4) = 3, etc. For each of the eight kinds of valid marriage, there is also a rule for determining the “skin group” of the children:

Father Mother Children
1 2 8
2 1 3
3 4 2
4 3 5
5 6 4
6 5 7
7 8 6
8 7 1

We can define two mathematical functions, the father-of function φ and the mother-of function μ, that map the “skin group” of a father or mother to the “skin group” that their children must have: φ(1) = 8, μ(1) = 3, φ(2) = 3, μ(2) = 8, etc.

This is all much clearer when displayed visually. In the diagram, two-part black arrows →→ indicate valid marriages. The arrows run from the “skin group” of the wife to the “skin group” of the husband. Red arrows run from each marriage arrow to the “skin group” of the children. Together, the arrows form an octagonal prism:

Following a single black arrow and then a red arrow () gives the mother-of function μ, with μ(1) = 3, etc. It can be seen that this function has a four-generation cycle: μ(μ(μ(μ(x)))) = x or, as it is often expressed, μ4(x) = x. In other words, each person’s “skin group” is the same as that of their great-great-grandmother in the female line (the maternal grandmother of their maternal grandmother).

Following a single black arrow backwards (from the head end) and then a red arrow () gives the father-of function φ, with φ(1) = 8, etc. It can be seen that this function has a two-generation cycle: φ(φ(x)) = x or, as it is often expressed, φ2(x) = x. In other words, each person’s “skin group” is the same as that of their grandfather in the male line (their paternal grandfather).

The combination of the two cycles makes the Lardil “skin group” system a very effective way of shuffling genes within a small population, thus avoiding inbreeding. It also highlights the fact that mathematics can be found in some surprising places. And there are even more patterns to be found in this example. Among others, σ(x) = φ(μ(x)). Also, μ(φ(μ(x))) = φ(x), which some readers may recognise as indicating a dihedral group.