Good Arrays

Discrete Mathematics Level 5

Consider a \(7 \times 7\) array, each of whose cells is filled up by an integer between \(1\) and \(49\) (inclusive). Every cell has exactly one number written on it, and every number between \(1\) and \(49\) appears exactly once.

An operation on the array is defined as follows:

  • Select a row / column of the array.
  • Either add \(1\) to all numbers in that row / column or subtract \(1\) from all numbers in that row / column.

An array is said to be good if there exists a finite sequence of operations after which all cells have the same number written on them. Find the last three digits of the number of good arrays.

Details and assumptions

  • A row / column can be operated on as many times as wanted.
  • You might use the fact that \(7\) is a prime.
  • This problem is not original.

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