The knight's tour

An old question

Players in India and the Arab world wrote out tours more than a thousand years ago. Euler studied them in 1759. A tour is closed when the last square is one knight's move from the first, and open otherwise. On an 8×8 board there are billions of tours, so finding one is easy for a computer. Finding one with a special property is not.

Beverley's tour, 1848

In 1848 William Beverley published a tour with a striking property: every row and every column of the numbered 8×8 board adds up to 260. That makes it semi-magic — magic in rows and columns, though not on both diagonals. It was the first tour known to do this, and such tours turned out to be rare.

How the sums work

Take a numbered tour, hide some of the numbers, and print beside each row and below each column the sum of the hidden numbers in that line. Each sum is a constraint: the missing numbers in a row must add up to it, and consecutive numbers must still sit one knight's move apart.

1. One. Pick the line with the fewest empty squares.
2. Two. Subtract what is already placed from its printed sum; the rest is shared by the empty squares.
3. Three. Combine that with where the knight can actually go. Usually one number fits — and the next line opens up.

The full rules

The Beverley–Funk Tour, 2026

Philippe Funk, who designed SUKEIMA in 2013, turned Beverley's property into a clue. A Funk's Tour — or Beverley–Funk Tour — is a puzzle made from the sums, and each one is checked by computer to have exactly one solution.

The strangest case is the empty board: no numbers at all, only the sums. A tour and its reverse usually give different sums, so the sums alone can point to a single answer. They only coincide when the tour is semi-magic — Beverley's property, seen from the other side. That is the Sunday puzzle.