68 votes
Accepted

Can you put L trominos to fill the figure?

Answer: Reasoning:
Magma's user avatar
  • 5,259
56 votes
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A new way to cut a pizza

This paper by Joel Haddley and Stephen Worsley answers a slightly different question - finding monohedral disc dissections where not all pieces touch the centre - but the results generally apply to ...
Jaap Scherphuis's user avatar
52 votes
Accepted

Tiling with T-tetrominos in gravity

TLDR: I'll fill the board and prove that the solution is unique. First, let's start by: I'll paint those green: Let's repeat those steps a few more times, using orange, blue, red and purple, in ...
Victor Stafusa - BozoNaCadeia's user avatar
49 votes
Accepted

A rectangular room has a floor tiled with tiles of two shapes: 1×4 and 2×2

The answer is: Suppose we colour the floor of the room under the tiles like so: extending up to the edge of the grid. Then: So:
boboquack's user avatar
  • 22k
49 votes
Accepted

Which country flags can you make in Tetris?

samm82's user avatar
  • 7,263
46 votes
Accepted

A colorful dodecahedron

Partial Answer: Solution: Other Solutions: Fun Stuff:
DqwertyC's user avatar
  • 8,196
35 votes

A new way to cut a pizza

This is a minor upgrade on @sybog64's answer: One way of thinking about it is to start with this configuration and then taking groups of 2 slices and rotating each group by 120°.
loopy walt's user avatar
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33 votes
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Mutilated chessboard

I believe this works as a short proof.
Tyler Seacrest's user avatar
30 votes

A new way to cut a pizza

I haven't found a perfect solution, my pieces are symmetrical but not identical
sybog64's user avatar
  • 401
29 votes
Accepted

One rectangle, indivisible

The best you can do is one with an area of 30 (5 x 6): Disproving smaller cases 2 x 2 and 2 x 3 2 x anything 3 x 3 3 x 4 3 x anything 4 x 4 4 x 5 4 x 6 5 x 5 So that's definitely not an ...
Dan Russell's user avatar
29 votes

Find a heptagon with mirror symmetry that can tile a flat plane

One possible way is to use ...
M Oehm's user avatar
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28 votes
Accepted

Tiling a square with right-angled triangles

Note first that That's which immediately suggests the overall shape of the thing. With apologies for the horrific ASCII art: where Here, have some slightly less horrific not-ASCII not-art:
Gareth McCaughan's user avatar
26 votes
Accepted

Which heptomino is it obvious can't tile the plane?

The one that
RobPratt's user avatar
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24 votes

One rectangle, indivisible

Here is a proof that 5x6 is the smallest possible rectangle. A rectangle of size $x$ by $y$ has $\frac{xy}{2}$ dominoes and $x+y-2$ potential lines. All of these lines must be blocked by at least one ...
f'''s user avatar
  • 33.7k
24 votes

Tiling a square with right-angled triangles

May I offer a more aesthetic tiling?
Daniel Mathias's user avatar
23 votes

Tiling with T-tetrominos in gravity

I think that this tiling is a valid Tetris stack:
M Oehm's user avatar
  • 60.6k
22 votes

Minimize 1×3 tiles on a 5×5 table to block any more 1×3 tiles

UPDATE 2: To put OP out of their misery find now at the very bottom of this post an answer to what they probably mean. UPDATE: OP has changed the rules, so this is no longer valid, but see bottom of ...
loopy walt's user avatar
  • 21.2k
21 votes
Accepted

Near-fill with 3x1 long triominos, how to do a different void square than the center square?

The trick to this puzzle is to: (And here are those tilings: the center was already given, and the rest are obtainable from these by rotation.)
Deusovi's user avatar
  • 146k
21 votes
Accepted

Polly O'Mino's Hexcellent Adventure

COMPLETED GRID The first step: Next: An important side note: Moving on: The top shaded region: Hopefully, finishing up:
Jeremy Dover's user avatar
  • 27.5k
20 votes
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Hexomino Puzzle

1) 2)
Halvard Hummel's user avatar
20 votes
Accepted

Filling the plane with two colors

FINITE PORTION OF ANSWER This can be extended infinitely in all directions - see my route to solving below for how. First a detour to explain how I made a tool (which competing answers could also ...
Steve's user avatar
  • 3,875
20 votes

How to fully tile an 8 by 8 square with Z-tetrominoes?

If reflections are not allowed: The figures below show why. If reflections are allowed: See figure below showing the top three rows of the 8x8 square.
ApexPolenta's user avatar
  • 2,690
20 votes
Accepted

Tiling a 5-by-5 bathroom with L-shaped triomino tiles

The missing square has to be one of these: Demonstration that any of these is possible:
Carmeister's user avatar
  • 2,205
19 votes

Covering a 15x15 grid with rectangles

Here is a proof of the lower bound of 13: Why it works (and how I found it):
mathlander's user avatar
  • 1,241
19 votes
Accepted

Gimme five (Pentomino puzzle)

First, focus on Now, move on to
Bubbler's user avatar
  • 13.8k
18 votes
Accepted

Cover a single cube with FIVE identical cube nets

Parcly Taxel's user avatar
  • 7,623
18 votes

How to fully tile an 8 by 8 square with Z-tetrominoes?

\begin{matrix} 9 &1 &9 &5 &5 &9 &1 &9 \\ 1 &-11 &-3 &-7 &-7 &-3 &-11 &1 \\ 9 &-3 &5 &1 &1 &5 &-3 &9 \\ 5 &-7 &...
RobPratt's user avatar
  • 13.7k
17 votes
Accepted

The Rectangle Puzzle

Here is a general solution for n>6. Explanation: Here is a proof of why there is no solution for n=6.
Jaap Scherphuis's user avatar
17 votes
Accepted

Tiling a hexagonal chessboard with "tribones"

Colour in all cells in the top horizontal row in Red, the second in Blue, then Green, Red, Blue and so on. ...
IanF1's user avatar
  • 2,134
17 votes
Accepted

A flag-packing problem

I think I have the answer: The first step is to determine the color of the square numbers. Next, let us try the Again, the flag could be in Knowing this, we get A few more When you now consider ...
Jens's user avatar
  • 8,830

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