# Dissecting a figure into three congruent parts in three different ways

Figure 1 is divided in 2 equal parts of same size and shape in 3 different ways Figure 2 is divided in 3 equal parts of same size and shape in 2 different ways Is it possible to find a figure that can be divided in 3 equal parts of same size and shape in precisely 3 different ways using polyominoes, polyamonds or polyhexes?

• Do all three shapes need to be distinct? Or is it okay if there are, say, two different dissections into P-pentominoes?
– Deusovi
Oct 27, 2021 at 21:41
• Do you mean exactly 3 different ways? As it is possible to divide an equilateral triangle into 3 equal shapes in far more than 3 different ways...
– Stiv
Oct 27, 2021 at 21:56
• Here, "Exactly 3 ways" is going to be very Difficult to Prove. In your Examples, we can not Prove that there are no other ways to Dissect. We can change the Puzzle to (1) "At least 3 ways, with Each Solution having 3 Equal Parts and these Parts are Unique to that Solution" or (2) "At least 3 ways, with Each Solution having 3 Equal Parts and these Parts are not necessarily Unique to that Solution" ....
– Prem
Oct 28, 2021 at 7:29
• If the three dissections did not need to use different shapes then example #2 could already be done in three ways: In the L-pentomino dissection the L-pair forming a rectangle can be flipped. Oct 28, 2021 at 9:27
• @Prem if we go out of polyamonds or polyominoes I accept any solution that has 3 or more different solutions but not infinity Oct 28, 2021 at 20:05

I'm still very unsure about the puzzle. But if I understand the rules correctly, does this answer your question in polyamonds?

• The first two dissections use the same shapes. Mirror images do not count as different. Oct 31, 2021 at 4:57
• @JaapScherphuis My reading comprehension might be bad, but I thought it's pointed several times from comments that mirror images are different? (I asked about L and J-tetrominoes cases as well although now I realized OP's answer is very vague for me...) Oct 31, 2021 at 5:58
• $\raise -.1ex@$athin, @Jaap Scherphuis and you are truly inspired dissecters. As for one of my dissection puzzles, you snuck up on the solution. My sniffer senses the same here. ^vote in imagination of what all solutions you've already considered and what you might yet.
– humn
Oct 31, 2021 at 9:06
• @athin in figure 1 and 2 you repeate pieces, is considered same solution Oct 31, 2021 at 18:02
• @RodolfoKurchan May I know what exactly does "repeating pieces" mean? Correct me if I'm wrong: so mirroring counts as one? Oct 31, 2021 at 23:44

Here is a heptiamond tiling with exactly three sets of three. Method of solution: Start with an arbitrary shape with 60 degree rotational symmetry, I used a triangle of side 6 which fits 3 12-iamonds. Fed it through my solver which says 20 12-iamonds tile it. Move on to smaller and smaller shapes and smaller sets (eg octiamonds into a side-2 hexagon has 5 tilings) until I settled on heptiamonds into a 2-hex which has 24 triangles, needing three removed. This was the first location I tried for those three. Oh and I tried polyominoes too, finding very few results up to 12-ominoes in a multitude of shapes.

Here's a triple for three tetrahexes...

• There are more than three ways to divide up that figure into three congruent pieces. I don't have the ability at the moment to draw it, but another way is the same as your first way except that the diagonal line joining the top of the right triangle to the top-right corner of the figure instead joins the top of the right triangle to the top-left corner of the figure (and the others are adjusted to match). (Note that the question didn't demand division along omino lines.) Oct 29, 2021 at 3:33
• Actually you can change the angle of that line by an arbitrary amount. I suspect that any shape based on 60 degree rotational symmetry will have this issue. And figures not based on such symmetry are much harder to find... back to the drawing board. Oct 29, 2021 at 3:43
• @theonetruepath your solution is OK for the part of polyamonds. It will be great if there is a solution with polyominoes, or someone can prove that it is impossible. Oct 29, 2021 at 4:35
• @theonetruepath can you find a solution for polyhexes? Oct 30, 2021 at 0:59
• @RodolfoKurchan Probably... my first attempt was a hexagon with area 19 cells, remove centre, then four different hexahexes tile it. You have the same sort of lattitude as polyiamonds and similar ways to deform each tiling into an infinity of shapes. Oct 30, 2021 at 3:43

Here is a solution with polyamonds of a figure that can be divided in 3 equal parts of same size and shape in precisely 3 different ways: • This is incorrect; the given shape can be divided in 3 equals parts of same size and shape in more than 3 different ways. For example, here is another way.
– Avi
Oct 28, 2021 at 20:24
• Excellent @Avi so the puzzle is still open for polyamonds too! Oct 28, 2021 at 22:39
• So this answer makes an incorrect claim? I suggest you should either edit it or delete it then. Oct 29, 2021 at 5:48
• Your Dissection in Green & the Dissection given by @Avi both have Mirror Image Dissections, giving a total of 6 Ways .... [[ I am not DownVoting your Question & your Answer, but I can see why some users may have wanted to DownVote .... Basically, there is a lack of focus, which you should take as constructive feedback ]] .... Your Puzzle is not originally about Polyamonds, but after lots of refinement, your answer is restricting to Polyamonds ; While restricting to Polyamonds or Polyominoes, naturally there can not be infinite solutions. It will be good to add all criteria in the Question.
– Prem
Oct 29, 2021 at 6:50
• Thanks @Prem I ask now only using polyominoes, polyamonds or polyhexes Oct 30, 2021 at 0:59