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Tours on a 4 x n playing board

Problem 237

Published on Saturday, 21st March 2009, 01:00 pm; Solved by 1045; Difficulty rating: 70%

Let T(n) be the number of tours over a 4 × n playing board such that:

  • The tour starts in the top left corner.
  • The tour consists of moves that are up, down, left, or right one square.
  • The tour visits each square exactly once.
  • The tour ends in the bottom left corner.

The diagram shows one tour over a 4 × 10 board:

T(10) is 2329. What is T(1012) modulo 108?