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Is this Tetris puzzle solvable?

It is impossible.
• 13.7k
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• 4,454
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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 ...
• 8,858
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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 ...
• 44.1k
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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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• 5,553

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°.
• 12.2k
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Mutilated chessboard

I believe this works as a short proof.
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A new way to cut a pizza

I haven't found a perfect solution, my pieces are symmetrical but not identical
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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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Find a heptagon with mirror symmetry that can tile a flat plane

One possible way is to use ...
• 48.4k

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 ...
• 33.3k
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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 ...
• 6,874
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How to ship the new Slurm 7-pack efficiently

I have a 14x14 (28 case) rectangle. It is completely symmetrical. Someone beat that.
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Tiling with T-tetrominos in gravity

I think that this tiling is a valid Tetris stack:
• 48.4k

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 ...
• 12.2k
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COMPLETED GRID The first step: Next: An important side note: Moving on: The top shaded region: Hopefully, finishing up:
• 23.9k
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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.)
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1) 2)
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Solution!
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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 ...
• 3,807

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 ...
• 405
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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.
• 44.1k
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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 ...
• 8,760
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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 ...

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 ...
• 12.5k
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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. ...
• 2,104
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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:
• 111k