This piece was prepared for the
International Mathematical Olympiad
(IMO) 2023.
It features multiple levels of mathematical puzzles
(see the unfolded sheets and
mathematical problems below).
At one level, its surface describes **11 solved mathematical problems**
(P1–P11) and **3 unsolved problems** (U1–U3), all around a
central mathematical theme of
**dissections**.
At another level, these problems are written in two different
**mathematical fonts** also about dissection:
the solved problems are written in our
Tetris font
and the unsolved problems are written in our
2-piece disconnected dissection font.
Thus each letter of each problem is itself a puzzle:
for each letter of the Tetris font, the goal is dissect the shape into
exactly the seven Tetris pieces
(one-sided tetrominoes);
while for each letter of the dissection font, the goal is to dissect the
shape into two sets of pixels that can be re-arranged
(translated/rotated/reflected) into a square.
These 52 font puzzles are featured on the other side of the unfolded sheet
(and the base underneath the sculpture).
Each mathematical problem is also written within the shape of a
tetromino
(Tetris piece),
and the problems themselves form an interesting tetromino packing
(considered in Problem P3).

Watch a 7-minute video about the making of this piece, thanks to Jane Street!

[0691] “**Solve Me**” (2023), Mi-Teintes paper, 11" × 14" × 15" high:

One side of the unfolded sheet features 52 font puzzles.
For each of the 26 letters of the
Tetris font
at the top, dissect each letter into exactly the seven Tetris pieces.
For each of the 26 letters of the
2-piece disconnected dissection font,
dissect the shape into two sets of *not necessarily connected* pixels
that form a square (by translation and/or rotation and/or reflection).

The other side of the unfolded sheet features 11 solved problems (P1–P11, written in the Tetris font) and 3 unsolved problems (U1–U3, written in the 2-piece disconnected dissection font). These mathematical problems are detailed below.

Here is the text of the mathematical problems, along with related hints and/or references:

**P** = puzzle / problem

**U** = unsolved / unknown

**P1** Give an efficient algorithm to tile a given polyomino by 2x2 squares if possible

**P2** Pack 2 complete sets of the 7 Tetris pieces into a 7x8 rectangle

AAEEJJJ AAEIIMJ BBEFIMM BBFFIMN CDFGKNN CDGGKKN CDGHHKL CDHHLLL

**P3** Show 2 complete sets of the 7 Tetris pieces cannot pack into [O]

**P4** Prove that a tiling of a square by rectangles with no separating line has 1 piece
strictly inside the square

**P5** Which tilings by Tetris pieces can be ordered so each piece is supported by a prior
pixel or the floor?

**P6** Prove NP-hardness of tiling a given polyomino with any trominoes

**U1** What is the complexity of tiling a given polyomino with any tetrominoes?

**P7** Cut the surface of a cube into 2 pieces that each fold into a unit cube? 4 pieces? 5
or 8 or 9 or 10 pieces?

**P8** Prove 2 equal-area rectangles have a common dissection into finitely many polygons

**P9** Assuming P8, prove 2 equal-area polygons have a common dissection
into finitely many polygons

**P10** Prove some 2 equal-area polygons have no k-piece common dissection, for any k

**U2** Do a square and a regular triangle have a 2-piece or 3-piece common dissection?

**P11** Prove NP-hardness of finding a k-piece common dissection of 2 given polygons? In the
strong sense?

**U3** Does any algorithm decide whether 2 given polygons have a 2-piece common dissection?
*k*-piece?