# 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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### 8x8 grid with no unmarked L-pentomino

What is the minimum number of cells on a 8x8 chessboard that need to be marked so that the unmarked cells do not contain an L-pentomino? An L-pentomino looks like ...
756 views

### Is it possible to arrange the free n-minoes of orders 2, 3, 4 and 5 into a rectangle?

To be explicit, the shapes pictured below, with reflections permitted. Can these be packed into a rectangle? This puzzle arose from discussion on r/mathmemes. No solution was posted (and I don't know ...
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### 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-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
402 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 ...
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### print("Hello, World!")

Build HELLO. Only rotations, no reflections.
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### 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.
315 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 ...
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### 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 ...
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### 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 ...
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### The Diamond of Columbus

There are up to flipping / rotation 12 distinct pentominoes. Can you fit them without overlap in the white area?
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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-...
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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 ...
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### 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 ...
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### 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:...
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### 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 ...
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### 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...)
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### 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 ...
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### 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 ...
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### 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 ...
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### "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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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### 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,...
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### 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 ...
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### 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 ...
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### 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,...
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### 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-...
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### Polyominoes inside a 10x10 grid

Can you place five dominoes, five trominoes, five tetrominoes, five pentominoes and five hexominoes inside a 10x10 grid, such that: No two polyominoes overlap No two polyominoes of the same size (by ...
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### Multi-colored polyominoes inside a 7x7 grid

Can you place four red trominoes, four green tetrominoes and four blue pentominoes inside an 7x7 grid, such that: No two polyominoes overlap No two polyominoes of the same color touch each other ...
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### Tetromino in a Pentomino Lair

Inspired by this question: Can you fit twelve pentominoes (not necessarily distinct) and one tetromino inside a 10 x 10 grid such that they do not overlap or touch each other orthogonally (...
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### 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 ...
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### Ten tetrominoes inside an 8x8 grid

Can you place ten tetrominoes inside an 8x8 grid, such that they do not overlap or touch each other orthogonally (horizontally or vertically) ?
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### 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 ...
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### How many distinct pentominos can be placed on a 8×9 board?

Upon proving optimality of an 8-pentomino solution for an 8×8 board, I was curious to see whether there is a 9-pentomino solution for an 8×9 board, namely a way to arrange 9 distinct pentominos within ...
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