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Vamshi Jandhyala

Mathematics

Chess Capture Digraphs

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Erich Friedman asked which directed graphs arise when an arrow runs from each chess piece to every piece it attacks. Here is a position for digraph u, which his answers page lists as having none, and smaller boards for digraphs h, k and l.

The rules

In his Math Magic column for October 2013, Erich Friedman asked which directed graphs can be drawn by chess pieces on a board, one piece for each point, following these rules.

  • The pieces are kings, queens, rooks, bishops and knights, with no pawns. Any number of each may be used.
  • Colour is ignored, so any piece can attack any other.
  • There is an arrow from one piece to another exactly when the first attacks the second: every attack is an arrow and every arrow is an attack.
  • A piece attacks another if it could move to that square in one move. So a rook, bishop or queen attacks only the first piece in each direction, and a king or knight cannot be blocked.
  • In these digraphs each piece attacks exactly two others and is attacked by exactly two.
  • The board may be any rectangle, and the aim is the smallest area.

In each figure below, the digraph is drawn as on Friedman’s page, with the piece and its square at each point.

Digraph u

Friedman’s answers page lists digraph u as having no position (as of 27 September 2026). It has one, on a board of twelve squares, and no smaller board works.

Left: digraph u, six points on a hexagon joined by twelve one-way arrows, each point showing a chess piece and its square. Right: a board three squares wide and four high with a knight on b4, a rook on b3, a bishop on c3, a king on a2, a rook on c2 and a knight on a1.
Digraph u on a 3 × 4 board.

Each piece attacks exactly two others. The rook on c2 attacks the bishop on c3 and the king on a2. The knight on b4 attacks the rook on c2 and the king. The bishop attacks both knights. The rook on b3 attacks the knight on b4 and the bishop. The knight on a1 attacks both rooks. The king attacks the rook on b3 and the knight on a1.

Smaller boards for h, k and l

Friedman’s page already has positions for digraphs h, k and l, on boards of 16, 15 and 16 squares. Each also fits on a 3 × 4 board.

Left: digraph h on a hexagon, with two-way arrows along the top, the bottom and across the middle. Right: a three by four board with a queen on a4, a king on a3, a knight on c3, a knight on a2, a king on c2 and a queen on c1.
Digraph h on a 3 × 4 board (the page shows 4 × 4).
Left: digraph k, five points on a pentagon around a centre point. Right: a three by four board with rooks on a4 and c4, kings on a2 and c2, a knight on b2 and a bishop on b1.
Digraph k on a 3 × 4 board (the page shows 5 × 3).
Left: digraph l, five points on a pentagon around a centre point. Right: a three by four board with a rook on b4, a king on c4, a rook on c3, a knight on c2, a queen on a1 and a king on b1.
Digraph l on a 3 × 4 board (the page shows 4 × 4).

The positions were found on 27 September 2026 with an AI-assisted search. Each was checked square by square against its digraph by a separate program, and the board for u against every smaller board.