# Knight's Tour

Knight's Tour is a type of path-drawing logic puzzle. The puzzle is played on a square grid usually containing some numbers, where the objective is to visit all the squares in the grid exactly once while always making a knight's move in chess, with the numbers providing a partial order on the squares. It can also be seen as a variation on Hidato and Numbrix, which have different movement rules.

## Background

### Rules

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Knight's Tour is played on a square grid usually containing some numbers. The basic rules are as follows:

• Write a different number from 1 - N in each cell of the grid, where N is the total number of squares.
• Adjacent numbers (that is, A and A+1) must be a knight's move apart in chess. That is, they must be two columns and one row apart, or two rows and one column apart.
• In Hidato, adjacent numbers must be a king's move (alternatively, the cells must at least share a corner). In Numbrix, adjacent numbers must be orthogonally adjacent (that is, the cells must share a side).

Additionally, there may be an additional condition that the last square is also a knight's move away from the starting square. Such a path may be considered to be closed, re-entrant, or circular.

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## Strategy

• It's useful to consider squares that have only two neighbors, as you can immediately get parts of the path, or the endpoints.
• Specifically for Knight's Tour and Numbrix, if you color the board like a checkerboard, then the path must alternate between colors. This parity restriction may provide some constraints for the path.
• It's also useful to consider numbers that are far apart spatially but have nearby numbers, as this also provides constraints on the path. For a Knight's Tour in particular, distance may be somewhat misleading though - two squares that are orthogonally adjacent will require 3 knight's moves, and two squares that are 2 squares diagonally away will require 4 knight's moves.

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## Notable Examples

• Night Vision Goggles (GPH 2020) (web) - This meta uses the Knight/night pun in several ways, including involving a Knight's Tour on a 6x6 grid.
• Continuity (Silph 2021) (web) - This is a Hidato puzzle that uses letters from A-Y placed into a 5x5 grid.
• Messenger (Microsoft 22) (web) - Click to revealIt's not immediately clear that this is a Hidato puzzle, although applying the alphanumeric A1Z26 on the given numbers gives the message HIDATO. This puzzle turns out to be a Hidato and TomTom hybrid.