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#39 Game of logic

set by Peter Rowlett

Lewis Carroll鈥檚 Game of Logic involves placing counters on a board to represent logical statements. The top half of the board is x and the bottom half is 鈥渘ot x鈥�, denoted by x鈥�. The left and right halves are y and y鈥�. The middle sections are m, and the outer sections are m鈥� (not labelled).

Say x means 鈥渘ew cake鈥�, y means 鈥渘ice cake鈥� and m means 鈥渨holesome cake鈥�. A red counter on the board means some cakes are in that section, and a grey counter means no cakes are in that section. The arrangement above says there are cakes that are new and nice, but none are wholesome. Where would you place a red counter to indicate there are some cakes that are new and wholesome, but not nice?

In this cake example, Carroll mentioned some 鈥渞emarkably untempting鈥� old cakes that are unwholesome and not nice. Can you place a counter to represent them?

Start with a fresh board and represent this statement: 鈥渟ome wholesome cakes are new, and no wholesome cakes are not nice鈥�. Are there nice cakes?

Solution next week

#38 Rotating dice

Solution

There are 24 ways to place the cube. You can put any of the six faces on top, with four ways to orient the side faces. A cube also has 13 axes of rotation, yielding 24 rotations: three axes through the centres of faces, each rotating by 90, 180 or 270 degrees; four through opposite corners, rotating two ways each; and six through the centres of opposite sides, each with one rotation. The non- rotation makes it 24. For a cube, and any regular solid, the number of rotations is equal to the number of ways it can be placed, and is 2n, where n is the number of edges. Any positioning of the cube can be achieved by some rotation, and for any given edge, there will be a rotation which moves that edge to another specific position.