# Find 7 solutions

You have three flat pieces, as shown:

Arrange them flat, without overlap, such that the shape formed by the black parts is congruent to the shape formed by the white parts. Rotation and reflection are allowed.

Find at least 7 distinct solutions.

If you find a single one, feel free to post a partial.

Quick note to clarify the aim of this puzzle: This is not a trick question where you need to stack the shapes/make a 3d shape etc. It's exactly what it appears to be. The solutions are just really hard to find.

• Either this needs a lateral thinking tag, or I didn't get the question or most likely I am plain stupid cause although the question says Find at least 7, I couldn't find a single 1 after 15 minutes. Feb 12, 2017 at 13:30
• @stackreader You're not stupid. It's just really hard. :P although you're right, it might need the lateral thinking tag. But that will give the wrong impression I think. Feb 12, 2017 at 13:37
• Does the shape formed by parts of a single colour have to be connected or can they be disjoint? Feb 12, 2017 at 13:43
• @sp3000 I don't see anything specifying they have to be connected :3 Feb 12, 2017 at 13:46
• I assume I can rotate the original pieces in an attempt to fit them together, can I also reflect them? Feb 13, 2017 at 16:03

8th one’s a charm, and might well be the sparsest solution without oblique rotations:

Equivalent independent earlier posts:
1. P.-S. Park
3, 4, 6. akhilesh

Solutions 7−12 have vector-like annotations, such as [-2B -1B 2x 2y ] for solution 12, because they can be generated formulaically.

Such configurations can be rotated 90° counter-clockwise to match single squares, A with A', and double squares, B with B' and C with C'. The orientation of the L-shaped piece is determined by ± 2B and ±1B while the S-shaped piece is oriented by ± 2x and ± 2y. This generates 16 configurations but some of them are invalid due to overlapped pieces. (In fact, [2B 1B 2x 2y ] produces an overlap and is inaccurately shown in the diagram.)

• Nice work, humn! You may notice that the real puzzle has been posted now :) Feb 12, 2017 at 21:58
• Great work with the formulation! Have you checked all the 16 configurations? If I get it right, $[-2_B, 1_B, 2_x, 2_y]$ is valid too, actually it gives the solution I listed below. I think another class of solutions with a 90-degree rotation and a reflection can be discussed with your method as well, but my brain starts to hurt seriously when I try to do that. Feb 15, 2017 at 16:48
• That's the vector for your partially connected solution alright, @elias, way to understand what isn't really explained! I was indeed wondering how you found that solution. Of the 11 configurations I checked, including yours, 3 have overlaps (unless one was a repeat). My brain began to hurt too in wondering if this formulation includes rotation in the other direction. At one point the straight piece had 4 orientations, for 64 total configurations, but half of those were definitely redundant, and all the discontiguous solutions found far fit this formulation.
– humn
Feb 15, 2017 at 17:03
• $[2_B, -1_B, 2_x, 2_y]$ seems to be valid as well, although I cannot see that among any answers posted here. Feb 15, 2017 at 17:06
• Yes, that's true, L and I pieces have to have parallel dominos for the 90-degree rotation, and because of that, the Z pieces have to be perpendicular to them (those are the results of their rotation). Feb 15, 2017 at 17:15

I found one.

XOO
OOXX
XXO


Is this one of 7 solutions?

Here's one, relying on the symmetry being allowed.

-xxxooo-
oo----xx

The following is courtesy of P.-S. Park and relies additionally on rotation. (Thanks!)

oo------
-xxxooo-
------xx

• You found 2 answers. The left block could be flipped upside down. Feb 12, 2017 at 16:24
• @P.-S.Park Brilliant! That's a rotation as well as a reflection. Feb 12, 2017 at 16:35

These 3 are possible solutions

• This answer deserves much more credit. All three of these solutions were posted before the accepted answer.
– humn
Feb 14, 2017 at 18:24
• A bounty is on its way
– humn
Feb 15, 2017 at 8:13

I haven't yet seen this one in the answers:

After humn posted a general formula about how to find those solutions which are based on a 90-degree rotation, I realised that my above answer could have been found with his method with parameters [-2B, 1B, 2x, 2y].

He may not have noticed, but he has found more solutions than he listed - both directly and indirectly. The above method with parameters [2B, -1B, 2x, 2y] provides a valid solution as well:

Just as with [-2B, -1B, -2x, -2y]:

And I think his method can be enhanced, at least to find those solutions which are based on a congruence that consists of a 90-degree rotation and a reflection parallel with one of the x- or y-axis - that is, a congruence which is a reflection to a reflection which has an axis that has a 45-degree angle with both the x- and y-axes.

• @humn, I've just picked another of those fruits you've left low-hanging. Feb 15, 2017 at 18:19
• Sweet! And if anyone (one of us, elias?) finds a solution with a reflection that isn't already represented by solutions here, it could even be posted as a new puzzle. Changing the Z piece to a tall-L piece might be interesting enough for a new puzzle too, perhaps simplified by limiting rotation to 90-degree multiples and by adding a goal such as most or least compact.
– humn
Feb 15, 2017 at 18:25
• @humn, I think I actually managed to formalise the other class, and in fact it does not lead to any solutions. Knowing this result I also managed to get an easy explanation. If you think about that transformation as a reflection to a 45-degree-axis, it is intuitive why: There are no pieces allowed to lie on the axis (they would be transformed into themselves, producing an overlap). But then you cannot distribute the given tiles in a manner that the number of pieces on the two sides of the axis are equal. Feb 15, 2017 at 18:55
• On a second thought this heuristic explanation does not work, as there is a translation allowed after the reflection. However, I still think my calculations are correct and there is no such solution. I will organise my thoughts and post the results. Of course other kind of solutions are still possible. Feb 15, 2017 at 22:20
• More power to you, elias! (I'm looking forward to exploring more corners/variations of this puzzle now that I posed another - you might find it interesting too - that had been simmering for a year but got its final push from enjoying this one.)
– humn
Feb 15, 2017 at 22:25

Not sure if I'm interpreting the 'congruence' correctly, but how about this?

One side shows the shape in its original black and white, the other shows it in terms of which piece it is.

This one I'm a bit more unsure of:

I found another and was speeding through graphing it only to discover it was Lawrence's. He beat me to the bridge! How many do we have in total now, like 5?

• I'm afraid neither of these pass the criteria about congruence. The black areas (to be transformed together!) cannot be transformed into the white ones using translation, rotation and reflection only. Feb 13, 2017 at 11:58