Combinatorics, Combinatorial Geometry, Grid Paper Geometry / Lattice Geometry

This involves geometric problems set on a grid of equally spaced points (a lattice), often where coordinates are integers. Questions might involve finding areas of polygons using grid points (e.g., Pick's Theorem), counting lattice points, or analyzing shapes drawn on grid paper.

• THE MAGIC KNIGHT'S TOUR

  Here is a problem that has never yet been solved, nor has its impossibility been demonstrated. Play the knight once to every square of the chessboard in a complete tour, numbering the squares in the order visited, so that when completed the square shall be "magic," adding up to `260` in every column, every row, and each of the two long diagonals. I shall give the best answer that I have been able to obtain, in which there is a slight error in the diagonals alone. Can a perfect solution be found? I am convinced that it cannot, but it is only a "pious opinion."

  Topics:

  Logic Combinatorics -> Number Tables Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 412