# Find 77777 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 77777 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.

• Shouldn't this be a computer-puzzle question? Feb 12, 2017 at 22:52
• Definitely not. I'll add no-computers now. With the right insight, finding 77777 solutions should be a breeze by hand. Feb 12, 2017 at 22:53
• @Beastlygerbil , if it makes things easier, try to find 777777777 solutions :P Feb 12, 2017 at 23:09

Here are 6 solutions, with instructions for another 77,771. The seed solution’s shape came from seeking threefold angular symmetry, for rotational simplicity, with the hope that one solution can become 77,777 merely by stretching such a configuration.   (Good news, stretching worked as hoped.)

Note that solutions 4− 6 are similar to solutions 1− 3, as if the pieces had passed through the center. Thanks to elias’s insight that one solution can become another by rotating the pieces by equal amounts, solutions 4− 6 can also be derived from solutions 1− 3 by turning each piece 180° locally before turning the entire plane 180°.

The seed solution’s details came from discovering that the pieces could be repeated to produce equal black and white patterns shifted relative to each other.

The pieces might as well have their own coordinate-system grids, at 120° angles to each other.   For any of 77,771 further solutions, just select virtually any coordinate pair and orientation and place each piece on its own grid, at those coordinates and with that orientation.   Here is a picture of those grids, with a solution for coordinates (-3­.­5, 4) and orientation 90° counter-clockwise and flipped.

Here are those pieces and their grids individually and compared to their seed positions /orientations.

These coordinate grids could also be extended out of the plane (violating the puzzle statement) to produce congruent 3-dimensional color patterns. That is, each piece could also be tilted and lifted out of the plane by a common arbitrary altitudinal angle and arbitrary offset along with elias’s arbitrary azimuthal angle.

• Nicely solved! 
– Deusovi
Feb 13, 2017 at 0:13
• Great job! Essentially all the solution variations are based on this same concept. Feb 13, 2017 at 0:35
• And you can be sure, @TheGreatEscaper, that the better picture(s) is (are) taking so long because it (they) include the negative case of this seed
– humn
Feb 13, 2017 at 0:37
• There's not just a negative case... you can characterise all infinite families in a simple way! Feb 13, 2017 at 0:49
• I think each piece can be rotated with the same - but arbitrary - $\varphi$ angle around its corresponding axis (marked by black dots on your drawings). Feb 13, 2017 at 6:30

How to make a solution in 3 easy steps! An illustrated guide to Congruence Infinity:

1. Drop a P pentomino (reflections allowed) somewhere on a plane, and also put a point somewhere (not too close to the P pentomino, please!)

1. Clone the P pentomino at 120 and 240 degrees rotated around the point

1. Place the three pieces on the three P pentominoes as follows (in any order):

Voila!

• Good for giving away your secrets! This is such a neat little set of pieces that pack into a couple of surprisingly difficult puzzles. I suspect some fun test-solving along the way.
– humn
Feb 13, 2017 at 13:13

Keeping the conditions in mind I came up with the following solution.

• what about the other 77776? Feb 13, 2017 at 8:20
• Good work! You might want to see this post, it has solutions which are non-infinite, including yours: puzzling.stackexchange.com/a/49018/30903 Feb 13, 2017 at 8:26