365 votes
Accepted

Is this Tetris puzzle solvable?

It is impossible.
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  • 13.7k
57 votes
Accepted

Can you put L trominos to fill the figure?

Answer: Reasoning:
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  • 4,454
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 ...
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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 ...
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22 votes

Tiling with T-tetrominos in gravity

I think that this tiling is a valid Tetris stack:
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  • 48.1k
21 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 ...
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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:
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  • 23.9k
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 ...
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  • 516
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.)
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  • 137k
20 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 ...
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19 votes
Accepted

Find smallest rectangle divided into figures so each figure has 5 neighbours

How's this? I got this by square-izing the icosahedron graph:
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  • 7,678
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 ...
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  • 3,807
19 votes

A Rook's Territory in the Chessboard

Here's an expandable solution for $n\ge 5$ (even or odd):
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  • 7,954
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 ...
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  • 1,073
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:
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  • 186
16 votes

Polyominoes to construct alphabet

Using:
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  • 35.4k
15 votes
Accepted

Dissecting Africa

Got the dissection part done. No idea about the native African word, though.
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15 votes

Tiling with Js and Ls

UPDATE Down to 6 outcolour tiles Here is one way:
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14 votes
Accepted

Polyomino T hexomino and rectangle packing into rectangle

2x2 - area 108 - optimal 2x3 - area 72 - optimal 2x4 - area 108 - optimal 3x4 - area 84 - optimal 1x5 - area 54 - optimal 2x5 - area 304 - optimal 3x5 - area 576 - optimal 2x6 - area 240 - ...
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  • 19.9k
13 votes
Accepted

Four Birds + One

Challenge 1: Fit the four yellow birds into the tray, no overlapping. Rotation and reflection are allowed. Challenge 2: Fit the four yellow birds and the blue piece into the tray, no overlapping. ...
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  • 1,110
13 votes
Accepted

How many colors does it take?

I think the answer is using the following coloring: For other board sizes, Reasoning:
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13 votes
Accepted

Generating all hexominoes by cutting and pasting

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  • 13.4k
12 votes

Find smallest rectangle divided into figures so each figure has 5 neighbours

Here is a proof that $12$ is the smallest possible number of regions in any feasible solution. Consider an arbitrary division of an arbitrary rectangle into $n$ regions, such that every region has ...
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  • 44.8k
12 votes
Accepted

Pairs of Pairs of Pentominoes

Finally arrived at this solution after playing around on https://www.scholastic.com/blueballiett/games/pentominoes_game.htm for way longer than I care to admit!!!! :) First pair (UI-TF) Second pair (...
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12 votes
Accepted

Tiling rectangles with W pentomino plus rectangles

First, a generalizable solution for $1 \times n$, $n$ is even. By halving the rectangles, we can also obtain solutions for odd $n$, and the parts with just rectangles and no W-pentominos can be ...
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12 votes

Covering an 8x8 grid with X pentominoes

Here's another proof of the lower bound in Sriotchilism O'Zaic's answer.
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  • 549
12 votes

What is the minimum-sized Blokus board which can contain all pieces?

Wow, that sure is tight. And yes, even though it doesn't look like it at the first glance, the I and T pentominoes are legally connected to each other :-) This took surprisingly long to find and (...
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  • 68.8k
11 votes
Accepted

Polyominoes on a checkerboard

The answer is Here's the proof, step by step (EDIT: the result was right, but the specific argument had a hole; I think I've patched that now) :
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11 votes
Accepted

Dissecting a square

I think I've found a solution for an 8x8 square. I do not know if it is the minimum solution or how to prove that: It was definitely fun to try and find this! Took me a while. Excellent puzzle. Some ...
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  • 17.8k
11 votes
Accepted

Pentominoes On the Edge

I found these solutions while playing around on https://www.scholastic.com/blueballiett/games/pentominoes_game.htm.
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