Mapping Checkmate Patterns with a 27-Dimensional Fingerprint

Some days ago, I read in a blog post on Lichess that the kill box checkmate pattern had been added as a puzzle keyword. This made me wonder how many checkmate patterns are without a specific name and “undiscovered”.

After all, we humans have a habit of naming patterns we find aesthetically pleasing or historically important, but there could be plenty more “nameless patterns” awaiting discovery.

The Kill Box

In the "Kill Box" checkmate, the Queen and Rook work together to trap and checkmate the King by forming a box. The Rook can move to c8, supported by the Queen, to deliver the final blow.

The squares surrounding the checkmated King, known as escape squares, are crucial in any checkmating pattern. To achieve checkmate, the attacker must control or eliminate all of these escape squares, ensuring the King has no way out.

The idea is to try to “measure” the 3x3 grid around the checkmated King. The parameters I have chosen to measure are pieces within the square (occ), attackers of squares (atk), and pinned defending pieces (pin).

Here’s how to read the 3×3×3 matrix—each square around the king lists three pieces of information in the format (occupant, attacker_sum, pin_flag):

1. Occupant – Which piece, if any, occupies that square.

   • A “0” means the square is empty.

   • A negative number indicates a defending piece (in this case, −6 = checkmate side King).

   • A positive number indicates an attacking piece (+4 = Rook).

   • “99” means that location is off the board (no real square).

2. Attacker_sum – The total “attacking power” from enemy pieces controlling that square.

   • Each piece type has a value (Pawn=1, Knight=2, Bishop=3, Rook=4, Queen=5, King=6).

   • If multiple pieces attack the same square, we add their values. So a square attacked by both a Rook and a Queen shows 4+5=9.

   • If no attacker targets that square, it’s 0.

3. pin_flag – Indicates whether a defending piece on this square is pinned.

   • “1” means the occupant (if negative) can’t legally move because it would expose its king.

   • “0” means not pinned (or no occupant at all).

3×3×3 matrix: (occupant, attacker_sum, pin_flag)

(0, 5, 0) | (0, 5, 0) | (0, 4, 0)
(0, 5, 0) | (-6, 4, 0) | (4, 5, 0)
(99, 0, 0) | (99, 0, 0) | (99, 0, 0)

Matrix visualization

When we combine the 9 squares with 3 values each we get a 27-dimensional fingerprint. The fingerprint represents the checkmate position with the embedded data about the “kill box” around the King.

Next I picked a dataset of 3000+ checkmate puzzles from the Lichess puzzle database and generated a fingerprint value for each checkmate. Once each has these 27-number fingerprints for all puzzles, it is possible to feed them into a clustering algorithm.

That algorithm looks at how “close” or “similar” each fingerprint is to others. Puzzles with similar fingerprints get grouped into the same cluster, meaning they share the same patterns of attackers, defenders, and pins around the king. By doing this for all your puzzles, it should be possible to discover families of checkmate patterns without having to label them.

The clustering of checkmate patterns

Out of the 3200 checkmate positions, around 1800 were fitted within a cluster. It was possible to create 74 pattern clusters. The interesting thing now is to investigate if the checkmate patterns have been ordered into patterns that can be interpreted by the human brain. For this, I made diagrams of the positions and ordered them after their clusters. The layout is a bit sloppy, but it does the job.

Cluster 9, 34, and 53 - Examples

Checkmates from cluster 34

This checkmate pattern is a version of the Bishop and knight mate. While cluster 53 is a kind of Epaulette mate in the open? You can try the top left puzzle (#1411, cluster: 53): here

A final example is cluster 9, which seems to have the corner square as the denominator.

I haven’t looked through all of the 74 detected checkmate patterns, but overall, I think this method of looking at the “kill box” around the King can be used to group checkmate puzzles into meaningful patterns.

I would be interested in hearing your thoughts about this idea. Maybe you have ideas for improving the matrix. It might be an interesting concept for a checkmate pattern puzzle book?

Enjoy the weekend!

/Martin