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These games are called "finite subtraction games" or "substraction games with a finite subtraction set". As Per Alexandersson mentioned, there are only $2^{a_k}$ sequences of 0s and 1s of a length that could affect the next $w_n$, so the sequence of 0s and 1s must be eventually periodic. There is actually a lot more that you can say. If you have heard of ...


Take the fractions $i/j$ in a regular, square array. Put each fraction in lowest terms, so 5/5 becomes 1/1. The number in the array is the sum of the numerator and denominator.


row numbers are increment by leaving 1 extra space in every row. like first row Second row 1.x.2.x3.... third row 1.x.x2.x.x.3... where as column number increments at the place of x. Row is working like this 1 2 3 4 5 6 7 8 9 1 x 2 x 3 x 4 x 5 1 x x 2 x x 3 x x 1 x x x 2 x x x 3 1 ...


Hint: The $8$th, $16$th, and $24$th shapes will be the circle at the end of the cycle, and are each followed by a triangle. Consider dividing by $8$. What do you remember about "remainders"?

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