Four fanatics and one checkerboard

Four checkers-playing fanatics eagerly pair up for two simultaneous games of checkers but somehow find themselves with just one board. They do have enough checkers for two games, so it is time to act. ­ One player hastily cuts that board into 9 pieces and reassembles them into two identical oddly-shaped boards with congruent rectangular “squares,” thus saving the day.

Yes, these are the same enthusiasts who once recklessly played a rash round robin on a board made of poison oak and should have learned their lesson: ­ Always bring more than one board.

Later they realize that their games could instead have been played on a different concoction of two identical separate boards, but consisting entirely of congruent true squares and being reassembled from a dissection into only 4 pieces.

Every possible checkers game would progress through exactly the same options — whether forced, taken or not — as on the original standard board.

How?

A correct solution will treat the original board as having a single 2-dimensional playing surface and use all of it with no overlaps, folds, gaps or additions. (Any interesting approaches that disregard these parameters are welcome, though, and deserve votes of approval. Three pieces might suffice, for instance, if the boards are not separate, or gaps could be used as squares, or “squares” might not even be congruent rectangles, or folding could allow for a single straight cut, or Rand al'Thor could slice a 3-dimensional board laterally, or . . .)

• We can do it in just 2 pieces if we're allowed to cut parallel to the plane of the board (and assuming the board is double-sided, or the square colours go all the way through). You might want to exclude that possibility :-) – Rand al'Thor Apr 5 '20 at 0:38
• Good point, @Rand al'Thor!. Better post your solution before i make your recommended edit and remove the lateral-dissecting tag. – humn Apr 5 '20 at 1:09
• Should the "congruent squares" are in $8 \times 8$ board? (Well.. If not, can we just divide the board to $2$ pieces and each will play in $4 \times 8$ hmm..) – athin Apr 5 '20 at 14:45
• Good thinking, @athin , how would the same game/setup/moves/options work on a $4 \times 8$ board? That question does relate to this puzzle's solution. Keep hmm-ing . . . – humn Apr 5 '20 at 15:44

You can cut the board to $$4$$ pieces like this:

And then reassemble the $$2$$ pieces like this:

Now you may wonder how to play the game with this new board. First, notice that:

The new board is actually representing the original board; specifically, the green cells here. (Try to look diagonally from lower-left to upper-right!)

To guide you:

Here is the starting position.

And the player will move as usual BUT in a horizontal/vertical manner instead of diagonally. The kings rows are on the top-leftmost and bottom-rightmost diagonals.

• @humn TBH, I was kept thinking how to divide the 8x8 cells into two boards (of 8x8 "something") which I was pretty sure it's impossible by 4 pieces dissection as rot13(rnpu pryy unf gb or qvivqrq vagb gjb cnegf). So that's why I clarified in the comment as the only possible solution will be rot13(guvegl gjb pryyf va rnpu arj obneqf) and suggested that 4x8 concept. The A-ha! moment is when you responded that the new board should be a Checker-playable-board. Once we figured out the "real" board by observing the piece's movement (my third spoiler above), the solution then becomes clearer. >< – athin Apr 6 '20 at 6:34
• That. Is. Brilliant. Hats off to you. – Rand al'Thor Apr 6 '20 at 11:20

I'm not sure why you'd need ANY sort of dissection for this.

You can put four players around the board, with two playing on the light squares, and two playing on the dark squares.

• Great lateral-thinking variation, @Braegh ! It got my instant ^vote for solving the fanatics' puzzle even if not this one. – humn Apr 5 '20 at 14:19
• This was literally my first thougt. :D – Vilx- Apr 6 '20 at 15:13

Is it that they:

Cut the board along its diagonals, create two squares while matching up colors, and play with their pieces on the corners of the new boards' squares?

• It's always good to include a picture. Apart from being a good way to illustrate your point. it serves as a sanity check for the answer. Here's one of the two boards you get with these instructions: i.imgur.com/eZUbhJo.png – Bass Apr 5 '20 at 10:48
• Thank you very much for the picture, @Bass . @AxiomaticSystem, I hope you don't my having edited the picture into your solution, but it does show why this isn't strictly correct as it includes triangles as well as squares. ("... with congruent true squares" in the puzzle meant "... consisting entirely of congruent squares," which is now clarified. I'm sorry if you took it as "... with two or more squares among other shapes") This solution got my instant ^vote, though, for its truly lateral-thinking way of placing the pieces! – humn Apr 5 '20 at 14:15