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Math Prelims question 1





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Infinite Chess

Consider an infinite two-dimensional grid of square cells which can be coloured either white or black (see the
interactive application). The white tiles represent the state 0 and the black tiles represent the state 1.
Start with a pattern of cells on the board (preferably in the centre) and play the animation. The initial
pattern evolves according to the following rule: at each time step, the colour of a cell is changed to the sum
of the colours of its four orthogonal neighbours, modulo 2.
In other words, the colour of a cell at each step is computed by taking the sum of the colours of the cells
situated at the top, bottom, left and right (with the convention that black cells represent 1 and white cells
represent 0) and colouring it white if the sum is even and black if it is odd. Note that all cells are updated
simultaneously.
Observe that while the initial pattern becomes blurred after the first few time-steps, copies of it eventually
re-emerge after a few more iterations of the animation. Explain this phenomenon.

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