# 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 18Andrew Bayly
 4 Pieces: period 8 period 12 period 14Joe DeVincentis period 16
 period 20Joe DeVincentis period 24Andrew Bayly period 40Joe DeVincentis
 5 Pieces: period 6 period 8Jon Palin period 10 period 12Jon Palin
 period 16 period 20 period 30 period 40George Sicherman period 70Joe DeVincentis
 6 Pieces: period 8 period 12 period 16Andrew Bayly period 20Joe DeVincentis
 period 24 period 36Jon Palin period 48Jon Palin period 60Joe DeVincentis
 period 72Jon Palin period 84Joe DeVincentis period 96Andrew Bayly
 7 Pieces: period 8 period 14 period 28Jon Palin period 56
 period 70Joe DeVincentis period 84Andrew Bayly period 98Joe DeVincentis period 112Andrew 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.