I love checkers - but I've played it so much that the color of my board is faded and dull. Today, I resolved to paint it. Unfortunately, I slipped and fell :(
Now there's paint all over my checkerpieces!
I only have 5 pieces left - the others fell down the couch somewhere. Could you help me paint my board? I want to paint the largest area possible with my 5 pieces.
- Each piece can only move once
- A move consists of hops over orthogonally (i.e. horizontal or vertical, not diagonal) adjacent pieces into empty squares.
- You can only hop over one piece at a time (you can't hop over two pieces in a row)
- Each move can consist of as many or as few hops as wanted and possible, and it is perfectly valid to double back on yourself.
- Once a piece is done moving, it stays on the board and can continue to be hopped over.
The goal is to paint as many squares of the checkerboard as possible. Every square that was ever in contact with a piece is considered painted - i.e. the starting square of a piece and every square that it hopped to.
You can choose the starting position of the pieces and the sequence of moves. (Two pieces can't start in the same position though)
This solution paints 8 squares using 4 pieces. I used black to represent unmoved pieces, and white to represent pieces that have already been moved. The numbers are there to help you understand the sequence of happenings. Note that the second piece hopped twice during its move, with the second hop being a return to its starting place.
I'll accept the answer that gets the farthest along in this list of criteria:
- Achieves the best solution
- Shows that their solution cannot be improved upon
- [Bonus] Generalizes their solution to arbitrarily many starting pieces - just in case I find the pieces that fell down the couch.
- You can assume the checkerboard is arbitrarily large.
- You do not need any calculations to do this, merely "elegant observations". (Even for the bonus problem)
- Checkers is normally played diagonally, but here pieces move orthogonally - this doesn't affect the puzzle. Can you see why?
- This problem originates from past me being bored during lockdown. I know the solution to the problems posed here, but I've not been able to solve it for most other grids (for example a hex grid) so if you enjoyed this puzzle and feel like setting a harder challenge for yourself, try doing it on a different grid.