366
votes
Accepted
57
votes
Accepted
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 ...
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 ...
48
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:
46
votes
Accepted
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°.
30
votes
Accepted
30
votes
A new way to cut a pizza
I haven't found a perfect solution, my pieces are symmetrical but not identical
28
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 ...
28
votes
Find a heptagon with mirror symmetry that can tile a flat plane
One possible way is to use ...
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 ...
23
votes
Accepted
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 ...
22
votes
Accepted
How to ship the new Slurm 7-pack efficiently
I have a 14x14 (28 case) rectangle. It is completely symmetrical. Someone beat that.
22
votes
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UPDATE: OP has changed the rules, so this is no longer valid, but see bottom of ...
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:
20
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.)
20
votes
Accepted
19
votes
Accepted
19
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 ...
17
votes
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 ...
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.
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
...
16
votes
Accepted
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
votes
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 ...
16
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.
...
16
votes
Accepted
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:
15
votes
Mosaic with tetris blocks
The Burr Tools freeware tool tells us there are exactly
How to use Burr Tools to solve this problem yourself:
The first tab is the Entities tab, where you define your shapes - both the puzzle ...
15
votes
Accepted
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