Easier way to understand:

See the whole thing as layers, like for cell x, layer 1 is the cell itself, layer 2 is cell x+1 and x-1 and so on. It is easy to see that this layer goes outward and from each layer another trajectory starts inwards. Let us say that somehow we know that for jumping to state y(true for the next state), we need to find XOR of cell x-y and x+y, so because of symmetry answer for state 2*y will be XOR of cell x-2*y and x+2*y...

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