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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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5 votes
2 answers
850 views

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 ...
Lucenaposition's user avatar
5 votes
3 answers
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 ...
ApexPolenta's user avatar
  • 3,168
12 votes
2 answers
481 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-...
bobble's user avatar
  • 10.3k
9 votes
1 answer
895 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 ...
Will.Octagon.Gibson's user avatar
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 ...
Bernardo Recamán Santos's user avatar
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 ...
Will.Octagon.Gibson's user avatar
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 ...
Lieutenant Zipp's user avatar
14 votes
1 answer
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 ...
username4231's user avatar
34 votes
1 answer
4k views

`print("Hello, World!")`

Build HELLO. Only rotations, no reflections.
Sny's user avatar
  • 3,185
7 votes
3 answers
728 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.
John Williams's user avatar
4 votes
1 answer
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 ...
Will.Octagon.Gibson's user avatar
8 votes
3 answers
483 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 ...
Will.Octagon.Gibson's user avatar
4 votes
2 answers
217 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 ...
Will.Octagon.Gibson's user avatar
3 votes
2 answers
712 views

The Diamond of Columbus

There are up to flipping / rotation 12 distinct pentominoes. Can you fit them without overlap in the white area?
loopy walt's user avatar
  • 21.3k
7 votes
1 answer
402 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 ...
Will.Octagon.Gibson's user avatar
13 votes
2 answers
490 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 ...
Rodolfo Kurchan's user avatar
10 votes
3 answers
865 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. ...
Dmitry Kamenetsky's user avatar
2 votes
3 answers
744 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 ...
Dmitry Kamenetsky's user avatar
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-...
Scratch---Cat's user avatar
18 votes
1 answer
606 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 ...
Jafe's user avatar
  • 78.7k
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 ...
Jafe's user avatar
  • 78.7k
8 votes
4 answers
895 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:...
Rodolfo Kurchan's user avatar
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 ...
Dmitry Kamenetsky's user avatar
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...)
Daniel's user avatar
  • 69
10 votes
1 answer
504 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 ...
Jafe's user avatar
  • 78.7k
6 votes
0 answers
229 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 ...
Maff's user avatar
  • 621
8 votes
2 answers
389 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 ...
Maff's user avatar
  • 621
9 votes
1 answer
397 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 ...
bobble's user avatar
  • 10.3k
9 votes
1 answer
603 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 ...
Jafe's user avatar
  • 78.7k
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 ...
Jafe's user avatar
  • 78.7k
3 votes
0 answers
224 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 ...
Bill's user avatar
  • 31
13 votes
4 answers
931 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 ...
justfortherec's user avatar
7 votes
1 answer
702 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 ...
Cheese Macken's user avatar
7 votes
1 answer
267 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 ...
Jafe's user avatar
  • 78.7k
8 votes
2 answers
359 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 ...
Jafe's user avatar
  • 78.7k
11 votes
1 answer
584 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 ...
Przemyslaw Remin's user avatar
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 ...
Florian F's user avatar
  • 30.8k
-2 votes
1 answer
163 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,...
Alain Reve's user avatar
3 votes
1 answer
225 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 ...
Slayveer's user avatar
2 votes
1 answer
127 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 ...
Will.Octagon.Gibson's user avatar
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,...
Will.Octagon.Gibson's user avatar
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-...
franck vivien's user avatar
4 votes
2 answers
419 views

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 ...
Dmitry Kamenetsky's user avatar
4 votes
3 answers
554 views

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 ...
Dmitry Kamenetsky's user avatar
8 votes
1 answer
227 views

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 (...
hexomino's user avatar
  • 138k
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 ...
Dmitry Kamenetsky's user avatar
10 votes
3 answers
1k views

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) ?
Dmitry Kamenetsky's user avatar
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 ...
mafu's user avatar
  • 299
3 votes
1 answer
359 views

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 ...
user21820's user avatar
  • 1,236
14 votes
3 answers
2k views

How many distinct pentominoes are possible to place on an 8 x 8 board?

Rules Place some pentominoes into an 8 x 8 grid. They do not touch each other. They can touch only diagonally (with corner). Pentominoes cannot repeat in the grid. Rotations and reflections of a ...
Ignac's user avatar
  • 141