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Questions tagged [polyomino]

A geometric puzzle centered around geometric figures formed from unit-squares, or a puzzle that uses polyominoes as an integral part.

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16 votes
7 answers
2k views

Covering an 8x8 grid with X pentominoes

What is the minimum number of X pentominoes you need to cover every cell of an 8x8 grid? Pentominoes may overlap each other and sit outside the boundary of the grid. An X pentomino looks like this:
12 votes
2 answers
462 views

A Crabby Sudoku

I present a *deep breath* Even-Odd TetroThermoDoku. From the back of the name to the front: Rules Sudoku: Fill each cell with a digit 1-9 such that no number repeats in any row, column, or 3x3 heavy-...
18 votes
4 answers
1k views

Tiling with Js and Ls

In this puzzle you must tile the plane with identically sized colored L and J tetraminos. To start I will place two of them like so: Your task will be to tile the entire rest of the plane meeting ...
9 votes
1 answer
884 views

Gimme five (Pentomino puzzle)

A pentomino is a tile made of five unit squares joined edge to edge. Divide this grid into five pentominoes, each containing the five letters A,B,C,D,E. The regions are not necessarily the same ...
26 votes
3 answers
3k views

Can you put L trominos to fill the figure?

In the above picture, there are 24 squares. Can you only use L trominos to fill the figure? If yes, give an example. Otherwise, please explain why. An L tromino is like this:
24 votes
2 answers
1k views

Hexominos from pentominos, heptominos from hexominos

All twelve pentominoes can be obtained by attaching a single unit square (edge to edge) to one of the squares that make up one (or more) of the following four tetrominoes: a) What is the least number ...
4 votes
3 answers
320 views

Tiling rectangles with U pentomino plus rectangles

Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with N pentomino plus rectangles ...
12 votes
3 answers
1k views

Tiling a 5-by-5 bathroom with L-shaped triomino tiles

The following puzzle appeared in the Cambridge, UK February 2024 MathsJam Shout: Tiling with Triominoes Imagine tiling a 5-by-5 bathroom with L-shaped triomino tiles. Clearly, not every square can be ...
12 votes
1 answer
1k views

Pentominoes On the Edge

Archived Source Introducing Pentominoes! It's the same concept as tetrominoes except they use 5 tiles instead of 4. Discounting rotations and reflections, there are 12 different free pentominoes. (If ...
17 votes
1 answer
1k views

Which heptomino is it obvious can't tile the plane?

A polyomino is a collection of equal-sized squares joined edge-to-edge in the plane (think Tetris pieces, but with an arbitrary number of squares instead of just four). A heptomino is a polyomino ...
14 votes
1 answer
392 views

Yet another pentomino puzzle

Just rearrange the 13 checkered polyominoes shown below to form a chessboard. The solution is unique and unusual. Clarification: The pieces may be reflected; the coloring on the back is as if the ink ...
34 votes
1 answer
4k views

`print("Hello, World!")`

Build HELLO. Only rotations, no reflections.
7 votes
3 answers
719 views

Making a 3 x 8 grid with tetrominoes

I'm wondering if it is possible to make a 3x8 grid using 6 tetrominoes: 2 Is, 2 Ls, a Z skew, and a square tetromino. I believe it may be impossible but would like to know why.
12 votes
3 answers
1k views

Dissecting a square

You are asked to dissect an $N \times N$ square into polyomino pieces such that each piece shares a portion of its boundary with exactly $D$ other pieces, and no piece has area exceeding $N$. This can ...
4 votes
1 answer
313 views

Xmas-colored tiles

If you subdivide a 2x2 tile into 4 unit squares and then color each unit square either red or green, then there are $2^4=16$ ways you can do this as shown below: Can you use all 16 tiles (rotations ...
14 votes
2 answers
1k views

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

The following 3x4 rectangle can be cut into pieces along grid lines, so that each piece has exactly three neighbors: Problem: Find the smallest rectangle on the integer grid that can be cut into ...
8 votes
3 answers
481 views

Ziggy - Can you make a square from 49 polyomino pieces

This puzzle is variant of the puzzle Ziggy - Make a square from 8 polyomino pieces by jaap-scherphuis. This puzzle is based on the fact that $1+2+3+\cdots+49=35^2$. It consists of 49 zigzag polyomino ...
4 votes
2 answers
214 views

Polyomino equation part 2

There is a particular shape (holes are permitted) that you can build with four copies of the piece on the left, or with 5 copies of the piece on the right. You may rotate and flip the pieces, but ...
3 votes
2 answers
711 views

The Diamond of Columbus

There are up to flipping / rotation 12 distinct pentominoes. Can you fit them without overlap in the white area?
7 votes
1 answer
399 views

Polyomino equation

There is a particular shape (holes are permitted) that you can build with three copies of the piece on the left, or with 5 copies of the piece on the right. You may rotate and flip the pieces, but ...
13 votes
2 answers
487 views

Minimum lines needed to draw the 12 pentominoes

You need 4 lines to draw the monomino and the same for the domino You can draw the 2 trominoes with 7 lines For the five tetrominoes I could do it first with 16 lines Then 15 lines And my best try ...
8 votes
4 answers
892 views

Smallest rectangle to put the 24 ABCD words combination

Put in the smallest possible size board all combination of 4 quantity of letters. Crossword must be connected. And can be only 4 letters words. Cannot be words with 2, 3, 5 or more letters Example for:...
10 votes
3 answers
863 views

Fewest polyominoes adjacent to 3 copies

What is the smallest positive number of polyominoes P, such that You can place grid aligned copies of P without any overlap; and Each polyomino is adjacent to exactly 3 other polyominoes. ...
2 votes
3 answers
742 views

Smallest polyomino adjacent to 3 copies

What is the smallest polyomino P in number of cells, such that You can place grid aligned copies of P without any overlap; and Each polyomino is adjacent to exactly 3 other polyominoes. Polyominoes ...
6 votes
1 answer
323 views

The Nine Gardens Of Eden

This puzzle is part of the Monthly Topic Challenge #11: Now in 3D. There are nine distinct known three-cell Garden of Eden patterns in the Life Without Death cellular automata, including the two well-...
18 votes
1 answer
602 views

Tetromino Jigsaw Madness

This is part 23 of the puzzle series Around the World in Many Days. Each part is solvable on its own. Dear Puzzling, This puzzle is a crossword-jigsaw hybrid. Cover the white area of the grid with ...
8 votes
1 answer
407 views

Restoring 3D Tetromino Puzzle

You won't believe this. I worked all night to make a new 3D tetromino puzzle, and just as I was about to save the final version there was a power outage! Now I can't remember what the final clues were ...
5 votes
3 answers
2k views

How to fully tile an 8 by 8 square with Z-tetrominoes?

Is it possible to fully tile an 8 by 8 square grid using only Z-tetrominoes? (I don't know the answer...)
13 votes
4 answers
908 views

What approach can I use to solve this wooden packing puzzle?

I am looking for an approach to solve a wooden packing puzzle which my three-year-old got as a present. We enthusiastically unpacked and disassembled the puzzle when she got it. (It came put-together ...
13 votes
1 answer
1k views

Sardines and Octopi

This is part 21 of the puzzle series Around the World in Many Days. Each part is solvable on its own. Dear Puzzling, The thick black lines in this grid form a LITS puzzle. Shade an orthogonally ...
17 votes
3 answers
2k views

Most polyominoes in an 8x8 grid

What is the most number of distinct free polyominoes you can form by painting an 8x8 grid in two colours? Here a polyomino is a set of orthogonally adjacent cells of the same colour, so polyominoes of ...
10 votes
1 answer
500 views

Coasting Along the Coast

This is part 14 of the puzzle series Around the World in Many Days. Each part is solvable on its own. Deаr Puzzling, The larger grid is a Fillomino puzzle. Each of the six different letters in the ...
6 votes
0 answers
227 views

Set of magic polyominoes that can tile a square

Let's first look at this square grid of numbers. The 9 squares in yellow is what we are looking at and the green numbers are the sums for the digits within the rows and columns. The red squares are ...
8 votes
2 answers
377 views

Polyominos packing into a square

Rules of the game: Take a square grid nxn. Populate the grid with polyonimoes of area size 1 to area of 9. Polyominoes can be any shape. There must be one each of every size polyomino in every row ...
20 votes
3 answers
2k views

The universal ticket

I am submitting a very interesting problem from a French mathematical recreation site: http://www.diophante.fr/problemes-par-themes/g-probabilites/g2-combinatoire-denombrements/1434-g248-le-billet-...
9 votes
1 answer
395 views

"Symmetric" Tetrikabe

This puzzle's theme is inspired by one from Grandmaster Puzzles' "The Art of Puzzles", but the puzzle itself is original. Rules: (Nurikabe section shamelessly stolen from an earlier puzzle ...
11 votes
3 answers
1k views

Fitting pentominoes inside a 10x10 grid

What is the most number of pentominoes that you can fit inside a 10x10 grid, such that they do not overlap or touch each other orthogonally (horizontally or vertically)? Bonus: what is the most number ...
11 votes
1 answer
1k views

How to solve this Pentomino puzzle?

The following Pentomino board has exactly one solution: The solution is known: I'm trying to solve this board manually, but it is so difficult that I came to believe it is impossible. I noticed ...
40 votes
6 answers
2k views

Packing pentominoes in a circle

You want to prepare a pizza of 12 flavors. You have 12 oddly-shaped pieces of cheese that you decide to use for the pizza. The shapes happen to be ... Oh, well, forget it! This isn't going to be ...
9 votes
1 answer
597 views

From Before Caesar

This is part 4 of the puzzle series Around the World in Many Days. Each part is solvable on its own. Deаr Puzzling, How have you been? I hope you don’t mind me showing up a little early this time ...
24 votes
1 answer
1k views

The woefully underclued crossword

This puzzle is part of the Monthly Topic Challenge #4: Cross-*non*-words Giving clues for everything makes things way too simple. In this crossword, four clues are not given and you must find a way ...
3 votes
0 answers
223 views

Cover a 15×8 board with the pentominoes [closed]

Cover the entire area of the 15×8 square shape with two complete sets of 12 different types of pentominoes. The pentominos can be rotated and flipped. Two pentominoes of the same type must not be ...
7 votes
1 answer
697 views

Solving a 5x5 pentomino with only certain shapes

I have a physical Pentomino puzzle lying around, which contains 2 F pieces, one Y piece, one T piece and one W piece. The area into which the squares are supposed to fit in is just a little under 6 ...
8 votes
2 answers
356 views

Shaka/shawaka: An introduction

This is a shaka/shawaka puzzle. Place black triangles in some white cells in the grid – formed by dividing a cell diagonally and painting one side black – so that each remaining white area forms the ...
7 votes
1 answer
266 views

Shaka/shawaka: Trominoes

This is a shaka/shawaka puzzle. Place black triangles in some white cells in the grid – formed by dividing a cell diagonally and painting one side black – so that each remaining white area forms the ...
8 votes
1 answer
513 views

Pentomino - is there any solution with the straight-bar piece in the middle of a rectangle?

Is there a solution for a straight-bar piece not touching the edge of the rectangle? By rectangle solution, I mean either one of the patterns 15x4, 12x5, or 10x6. By straight-bar piece, I mean the ...
-2 votes
1 answer
161 views

15 x 15 polyomino

You cannot move the red squares You cannot rotate the blocks You cannot have 2 block of the same color touching each other, not even diagonally (by their corners) Grey blocks cannot touch a red square,...
3 votes
1 answer
223 views

Block fill in date puzzle

I have been trying this puzzle for HOURS!!! The goal is the fit all the pieces but not cover Aug or 1. You can rotate the pieces. Credit to www.dragonfjord.com The link shown at the bottom of the ...
2 votes
1 answer
126 views

Tiling with dominoes then with tetrominoes

For reference, here are the five tetrominoes: Suppose you join ten dominoes to make a polyomino made of twenty unit squares. Is it possible that you can tile the polyomino using all the five given ...
8 votes
1 answer
515 views

Tiling a 5x5 square with five pentominoes AND a Latin square

Suppose that a 5x5 square has been tiled with five (not necessarily distinct) pentominoes. Is it true that there will necessarily exist at least one Latin square of size 5x5 (using the numbers 1,2,3,...