# The Magical Maze: Seeing the World Through Mathematical Eyes by Ian Stewart

By Ian Stewart

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Additional info for The Magical Maze: Seeing the World Through Mathematical Eyes

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Then we formed a graph whose nodes correspond to states and whose edges correspond to legal moves. The solution of the puzzle is then a path through the graph that joins the start to the finish. Such a path is usually obvious to the eye, provided the puzzle is sufficiently simple for the entire graph to be drawn. Puzzles of this type are really mazes, for a maze is just a graph drawn in a slightly different fashion. Metaphorically, they are logical mazes - you have to find the right sequence of moves to solve them.

So the golden angle is the one that gets as far away as possible from forming radial spokes. It is hardly surprising that it goes to the other extreme: dense packing. More recent work still, by the French physicists Yves Couder and Stefan Douady, has shown that this choice of angle is a natural consequence of the dynamics of a growing plant shoot . Each new primordium gets pushed into the biggest space available. That means that they all pack together efficiently, and that in turn implies that the golden angle is the most likely choice.

In column 4 the first and second entries must be different (panther and pig not left on same side when porridge is taken across). Now we can systematically complete the table for all the other seven states (Table 5). 1, 1) (1, 1, 1) (1, 1, 1) (0, 1, 1) (g, 0, 1) (,-,01) (0,1, -) (1,1,0) 41 So in fact, out of the 24 moves listed, precisely 8 are illegal and 16 legal. The 16 legal moves come in pairs: in each pair the same item is carried across the river in either direction. Thus we can move from (0,1, 0) to (1, 1, 0) or from (1, 1, 0) to (0, 1, 0).