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Consider a sequence of inscribed squares constructed on an $n \times n$ grid.

The grid has the following coordinate system:

  1. The unit square in the lower left corner has coordinates $(1, 1)$, and the upper right unit square has coordinates $(n, n)$.
  2. The first coordinate corresponds to horizontal shifts, and the second coordinate corresponds to vertical shifts.

All sequences have the following construction:

  1. The sequence starts from an $n \times n$ square.
  2. The next element in the sequence is an $(n - 1) \times (n - 1)$ square inscribed in the previous square.
  3. The last element of the sequence (unit square) is located in the square with coordinates $(x, y)$.

What is the general number of such sequences given the initial grid size ($n$) and the terminal element position $(x, y)$?

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  • $\begingroup$ Can you solve the analogous one-dimensional version? The posted version is just two copies of the one-dimensional version happening simultaneously. $\endgroup$ Commented May 25 at 15:10

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