Problem of the Month (April 2013)

Say we have a chessboard filled with white chess pieces and a cyclical order in which the pieces are to be moved. In turn, we move each piece as far as it can be moved (measured in number of squares), as long as that square is unique. (If no move is possible, or the longest move is not unique, we stop.) Some positions eventually repeat themselves. It is not hard to find positions where each piece moves only twice each period, or crowded positions where the pieces have to cycle around. It is harder to find more interesting examples.

What positions can you find that repeat? What periods are possible with a given number of pieces? What are the smallest boards (in terms of area) that accomplish this? Positions like this that contain kings and knights are harder to find, because of all of their moves are the same length. What positions using kings or knights can you find? What periods are possible with those pieces? What combinations of pieces have repeating positions?


ANSWERS

Here are the known periods 7 or fewer pieces, on the smallest-known boards:

1 Piece:
period 2
2 Pieces:
period 4
3 Pieces:
period 4

period 6

period 10

period 12

period 18
Andrew Bayly
4 Pieces:
period 8

period 12

period 14
Joe DeVincentis

period 16

period 20
Joe DeVincentis

period 24
Andrew Bayly

period 40
Joe DeVincentis
5 Pieces:
period 6

period 8
Jon Palin

period 10

period 12
Jon Palin

period 16

period 20

period 30

period 40
George Sicherman

period 70
Joe DeVincentis
6 Pieces:
period 8

period 12

period 16
Andrew Bayly

period 20
Joe DeVincentis

period 24

period 36
Jon Palin

period 48
Jon Palin

period 60
Joe DeVincentis

period 72
Jon Palin

period 84
Joe DeVincentis

period 96
Andrew Bayly
7 Pieces:
period 8

period 14

period 28
Jon Palin

period 56

period 70
Joe DeVincentis

period 84
Andrew Bayly

period 98
Joe DeVincentis

period 112
Andrew Bayly


All the known periods are even. Bryce Herdt asks whether there are any odd periods.

Here are the known periods using 3 or fewer pieces:

PiecesPeriodRectangles
2n×m, m≤3n-4
21×2
22×3, 2×4, 2×5
2m×n, except 1×1 and 2×2 (JD)
2m×n, except 1×1 and 2×2 (JD)
4n×m, n≤m≤3n-4
none
42×3, 2×4
4n×m, n≤m≤3n-4
4n×m, n≤m≤3n-4
none
none
none
none
42×3, 2×4, 2×5
42×4, 2×5, 2×6
42×4, 2×5, 2×6
4m×n, 2≤m≤n, except 2×2 (JD)
4m×n, 2≤m≤n, except 2×2 (JD)
4m×n, 2≤m≤n, except 2×2 (JD)
PiecesPeriodRectangles
4
6
10
(2n+1)×(2n+1), except 1×1
3×4, n×m, 4≤n≤m≤3n-5 (JD)
n×(3n-6), n≥5 (JD)
none
62×3 (JP)
6n×m, 3≤n≤m≤4n-4 (JD)
6
12
2×3, n×m, 3≤n≤m≤4n-4 (JD)
4≤n≤m≤3n-5 (JD)
none
none
none
none
62×4 (JP)
63×3 (JP), 2×5, 2×6 (JD)
62×5 (JP), 3×3 (AB), 2×4, 2×6 (JD)
63×3, 3×5, 3×6, n×m, 4≤n≤m≤3n-2 (JD)
6
12
3×3, 3×5, 3×6, n×m, 4≤n≤m≤4n-5 (JD)
7≤n≤m≤2n-6 (JD)
6
12
3×3, 3×5, 3×6, n×m, 4≤n≤m≤4n-5 (JD)
7≤n≤m≤2n-6 (JD)
PiecesPeriodRectangles
42×2
none
122×2
122×2
none
none
none
122×2
122×2
122×2
62×4 (JP)
6
18
2×6, 2×7, 2×8 (JD)
2×5
6
18
2×3, 2×4, 2×5 (JP), 2×6, 2×7, 2×8 (JD)
2×5
63×3 (JP)
6
12
3×3 (JD)
2×6 (JD)
122×6 (JD)
4
6
10
n×m, 2≤n≤m≤3n-4 (JD)
n×m, 3≤n≤m (AB)
2×n, n≥4 (JD)
6
12
n×m, 3≤n≤m (AB), 2×n, n≥4 (JD)
n×n, except 1×1
6
12
n×m, 3≤n≤m (AB), 2×n, n≥4 (JD)
n×n, except 1×1
4
6
n×n, except 1×1
n×m, 3≤n≤m (AB), 2×n, n≥4 (JD)

JP = Jon Palin
JD = Joe DeVincentis
AB = Andrew Bayly

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/18/13.