68
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 ...
- 9,431
46
votes
Accepted
39
votes
Hexominos from pentominos, heptominos from hexominos
Let us start by considering this hexomino:
It is clear that there is only one pentomino that can be extended to this:
And since we have to use that pentomino, we can tick off several hexominoes that ...
- 2,046
26
votes
Accepted
25
votes
Accepted
Now You're Packing with Portals #1
This works! (I think)
(Hopefully that’s clear enough how the shapes go)
I got this mostly by thinking about how the blue and red can be placed such that the top and bottom of the green can be ...
- 60.6k
23
votes
23
votes
Accepted
A Rook's Territory in the Chessboard
I started with this:
Pushed things this way and that, ended up with this:
Similarly, on 9x9:
And on 10x10:
It took me a while to get there, but that one suggests an emerging pattern.
And here is ...
- 15.4k
22
votes
Covering an 8x8 grid with X pentominoes
The X-pentomino tiles the plane, so that tiling is a good way to start. There are two ways to cut an 8x8 region out of that tiling. If one of the 4 central squares of the 8x8 region has an X centred ...
- 53.9k
22
votes
A Rook's Territory in the Chessboard
Here's an expandable solution for $n\ge 5$ (even or odd):
- 14.3k
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.)
- 148k
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:
- 28.8k
21
votes
How many distinct pentominoes are possible to place on an 8 x 8 board?
With integer programming, I managed to place
like this.
Here is my formulation. I happened to solve a similar model to solve a puzzle called One puzzle a day. Let $B$ be the set of cells in the 8x8 ...
- 525
21
votes
Accepted
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 ...
- 3,885
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:
- 2,547
19
votes
How many distinct pentominoes are possible to place on an 8 x 8 board?
I solved this completely by hand.
Here is a clean proof of its optimality. No computer is needed. Mere pencil and paper suffice.
Expand each pentomino by adding little right-angled isosceles ...
- 1,236
19
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.
- 3,168
19
votes
Accepted
18
votes
Accepted
Packing pentominoes in a circle
UPDATE 2
A minor improvement. New best radius
Arrangement
/UPDATE 2
UPDATE
New best radius:
using arrangement
/UPDATE
I get a radius of about
using the following scheme
which is obviously ...
- 21.3k
17
votes
Accepted
Polyominoes to construct alphabet
I haven't tried one of these before; I just stumbled across the question by accident. But I think I have an answer. You can build all 26 letters if you have a set containing:
Image below:
- 186
17
votes
The universal ticket
The previously best-known solution has score of 165, with the following grid:
From a clever brute-force search, one can learn that
However, you can do better! The ticket
achieves a score of
- 1,207
17
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 &...
- 14.3k
16
votes
16
votes
16
votes
Smallest rectangle to put the 24 ABCD words combination
Pretty sure that the following is minimal.
- 261
16
votes
Accepted
8x8 grid with no unmarked L-pentomino
I think the answer is
cells. One way to achieve it is as follows:
There is a straightforward proof that this is the minimum possible.
Bonus for some other pentominoes:
- 16.3k
15
votes
Accepted
Pentominoes On the Edge
I found these solutions while playing around on https://www.scholastic.com/blueballiett/games/pentominoes_game.htm.
- 2,511
15
votes
The universal ticket
Very unlikely to be optimal, but got to 120 on my first go:
Approach:
mess around with the problem until it becomes clear that connectivity of the squares will be the main problem.
invent glue, ...
- 77.9k
15
votes
Accepted
The woefully underclued crossword
Answers to clues:
At this point, given the tag, it looks like we are looking for
So let's start filling the grid:
Completed grid:
- 27.7k
Only top scored, non community-wiki answers of a minimum length are eligible