Problem 447. swap sign sum & multiply castles

  • It is an easy problem, if you know the answer.
  • Given a square matrix of NxN ordinary numbers.
  • Initially place N identical indistinguishable castles or rooks (chess pieces) on the main diagonal.
  • Then keep swapping any two rows or columns to exhaustively enumerate all possible unique patterns of castle formation.
  • Not a single castle in any of these formations should be under threat of any other castle,
  • only one castle watches over an otherwise empty row and column.
  • For each pattern, find the product of all numbers covered by the castles.
  • If this pattern was obtained after even number (0,2,4,...) of swaps,
  • then add the product to an initially empty accumulator,
  • otherwise subtract the product from the accumulator.
  • Give the final expected value of the accumulator,
  • does not matter whether by hook or by crook,
  • but please give a general solution,
  • the test suite may be modified soon.
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Solution Stats

51.72% Correct | 48.28% Incorrect
Last Solution submitted on Oct 02, 2025

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arithmetic chess matrix

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