372
votes
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61
votes
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52
votes
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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 ...
52
votes
Accepted
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 ...
- 47.3k
49
votes
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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:
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48
votes
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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°.
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31
votes
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30
votes
A new way to cut a pizza
I haven't found a perfect solution, my pieces are symmetrical but not identical
- 401
28
votes
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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 ...
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28
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Find a heptagon with mirror symmetry that can tile a flat plane
One possible way is to use ...
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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 ...
- 33.4k
23
votes
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Tiling a Chessboard with tetrominos
It is not possible. The area of a $10 \times 10$ checkerboard is $100$, so it takes $25$ T pieces to have the same area. The checkerboard has the same number of red and black squares, but each piece ...
- 7,146
22
votes
22
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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 ...
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21
votes
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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:
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20
votes
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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.)
- 143k
20
votes
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19
votes
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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 ...
- 3,815
18
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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.
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Occupy a field with tetrominos
Here is yet another solution with 9 pieces. This one is nice and symmetrical.
I have been trying to think of a way to show that 8 will not work by arguing in terms of the number of edge squares that ...
- 415
17
votes
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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.
- 47.3k
17
votes
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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.
...
- 2,124
17
votes
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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
...
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votes
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16
votes
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Tiling a Hexagon with Diamonds
I think I've found a really easy proof.
Every tile with vertical sides needs to have two other tiles with vertical sides adjacent to it, or the vertical boundary of the hexagon. For a given tile with ...
16
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Tiling a Hexagon with Diamonds
I want to post an answer that is more intuitive than mathematical.
This picture perfectly represents it:
White, grey and black are used to highlight the diamonds with the same orientation.
The right ...
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16
votes
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A chessboard tiling with corners removed in 3D
Our mutilated cube
Let's first consider
What about
Finally we must consider
So the final answer is:
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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 &...
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15
votes
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Only top scored, non community-wiki answers of a minimum length are eligible