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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A new beetle for 2014

Rentonium bicolor

The beetle above, from the North Island of New Zealand, is Rentonium bicolor. This colourful fungus-eating insect was described in the recent paper “A new species of mycophagous Rentonium (Coleoptera: Cleroidea: Trogossitidae) based on larvae and adults, and a catalogue of Rentoniinae” by Matthew Gimmel and New Zealand researcher Richard Leschen.

Who knows how many such insects are still waiting to be discovered?

Parasites and food webs


Parasitic copepod on the gill of a whiting (photo: Hans Hillewaert)

A recent paper “Parasites Affect Food Web Structure Primarily through Increased Diversity and Complexity” in PLoS Biology by Jennifer Dunne and others (including Robert Poulin from Otago) examines the effect of including parasites in food webs. The primary impact appears “attributable to the generic effects of increases in diversity and complexity, regardless of the identity or type of species and links being added.” However, parasites do impact network motifs because of, for example, the ingestion of parasites by predators of their hosts.

Fascinating work!


The Otago Harbour intertidal mudflat food web, with parasites shown in red

See also Assembling my network, Roopnarine’s foodweblog and the Parasite ecology blog.

Types of Influenza

Prompted by recent news about the H7N9 “bird flu,” the talented team at “Information is Beautiful” produced the graphic above (click on the image to zoom), based on this data. See also this press briefing from the CDC on H7N9. New Zealand readers will be interested in this graphic on influenza in Auckland:

Cubic Symmetric Graphs

The so-called “Foster Census” of cubic symmetric graphs (networks) has been very ably continued by New Zealand mathematician Professor Marston Conder. In August last year, he gave us a complete list of cubic symmetric graphs on up to 10,000 vertices. The chart above summarises that list, noting the diameter, girth, and order (number of vertices) of each graph. Each coloured dot is a graph. Seven of the better-known graphs are labelled, and the Tutte–Coxeter graph is illustrated (click on the chart for a larger image).


The University of Auckland: it’s not just fantastic movies that come out of New Zealand!