A 'modular knight' moves on circular chess board made from
concentric circles divided into sectors.
As a default, the board is split into 5 sectors with 2 concentric
tracks and the knight can move 3 steps forward (in any direction)
followed by 1 step to the side (in either direction), as shown in
the interactivity below. The middle and edge of the board are
joined so that when the knight moves over the outside edge of the
circular board it re-enters in the same sector on the inside of the
track (and vice versa).
Start the interactivity below by clicking on the +. The brown
squares represent the squares the knight has visited and the peach
squares the possible destinations on the next move.
To begin with, understand why all of the peach squares are
Then, can you make the knight visit every square once and only once
and return to its starting point?
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Suppose there are p sectors and q concentric tracks and a
knight's move is a steps in one direction and b steps in the other
direction. Find conditions on the numbers p, q, a and b under which
it is possible for the knight to visit every square and return to
its starting point.
Note that on the interactivity you can change the size of the
track, the direction and number of steps the knight can move
forward, and the number of steps it can move to the side.