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?
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 
Here are the known periods using 3 or fewer pieces:



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