Questions tagged [polyomino]

A geometric puzzle centered around geometric figures formed from unit-squares.

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15
votes
2answers
2k views

A Rook's Territory in the Chessboard

Shade 32 cells, four on each row and column, of an 8 x 8 blank chessboard (all its cells originally white) so that a rook sitting on any shaded cell can reach any other shaded cell, moving just along ...
8
votes
3answers
571 views

Tiling rectangles with F pentomino plus rectangles

Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with N pentomino plus rectangles Tiling rectangles with T pentomino plus rectangles ...
7
votes
1answer
216 views

Largest set of independent hexominoes

What is the largest set of hexominoes that can be found in which no two of them are such that one can be converted into the other by cutting out one of its component squares (thus obtaining a ...
7
votes
2answers
324 views

Finding the most flexible of all 35 hexominoes

Of all 35 hexominoes, which is (or are) the most flexible of all, that is, the one (or ones) that can be converted into the most other hexominoes just by cutting out one of its component squares (thus ...
12
votes
2answers
259 views

Let's Play Tetris!

Fill the whole grid with Tetrominos. Every cell must be part of exactly one Tetromino. Tetrominos can be rotated as necessary. The colored clue cells must be part of the indicated type of ...
11
votes
1answer
374 views

Generating all hexominoes by cutting and pasting

Place the 35 hexominoes around a circle in such a way that if two of the hexominoes find themselves next to each other, it is because one of the two can be obtained from the other by cutting out one ...
15
votes
2answers
762 views

Polyominoes to construct alphabet

It is possible, using a set of just 10 polyominoes, to construct any one of the 26 letters below. Can you find such a set? When constructing, polyominoes may be rotated and flipped, but may not ...
7
votes
1answer
108 views

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

Rules: Each player has the twelve pentominoes, five tetrominoes, two trominoes, one domino, and lone square. These may be flipped and rotated in any manner. The board on which these are to be placed ...
12
votes
3answers
381 views

Fewest polyominoes to construct digits

What is the fewest polyominoes you need so that any one of the numbers $0$ to $9$ can be constructed? When constructing, polyominoes may be rotated and flipped, but may not overlap. Bonus: How few ...
20
votes
3answers
605 views

Filling the plane with two colors

In this puzzle you must tile the plane with colored T-tetraminos. I will start by laying down 3 of them for you like so: Your task will be to tile the entire rest of the plane meeting the following ...
14
votes
6answers
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:
20
votes
3answers
2k 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:
8
votes
2answers
287 views

Hand tiling puzzle demonstrating Eisenstein triple $c^2 = a^2 -ab + b^2$

An Eisenstein triple is related to 60 degree triangles and a special case of the cosine law. But we need not worry about that except to note that a specific example of an Eisenstein triple is $7^2 = 5^...
4
votes
2answers
187 views

Covering an 8x8 grid with W pentominoes

What is the minimum number of W pentominoes you need to cover every cell of an 8x8 grid? Pentominoes may overlap each other and sit outside the boundary of the grid. They can also be rotated in any ...
0
votes
4answers
148 views

Rectangles formed from every tetromino, tromino and domino

Can you form a 4x7 rectangle from every tetromino, tromino and domino? There are 5 different tetrominoes, 2 trominoes and 1 domino. Can you find different arrangements that are not mirrors/rotations ...
10
votes
3answers
304 views

Four hand tiled squares demonstrating a Pythagorean Quadruple

Demonstrating the Pythagorean Quadruple $6\times6 + 6\times6 + 7\times7 = 11\times11$ Using the pieces shown in the $11\times11$ square: The objective: Arrange the pink pieces (four enneominoes) ...
46
votes
3answers
3k views

Tiling with T-tetrominos in gravity

The goal is to tile all the white squares using T-tetrominos when there is gravity pulling the tetrominos downwards like regular tetris. The black squares are void and the ground is just below the ...
250
votes
1answer
26k views

Is this Tetris puzzle solvable?

As a birthday present last year, I received some fridge magnets. They didn't come as a puzzle, so I don't know if they have a solution, but I made a puzzle out of them anyway. The magnets are ...
6
votes
1answer
215 views

Piece de Resistance - Eight Doubled Tetrominoes Make a Tetronogram

Eight Doubled Tetrominoes Make a Tetronogram This puzzle is part of the "Piece de Resistance" series. Go back to Part 1 (Ace) for the story.Ace Two Three Four Five Six Seven Eight ... Time for ...
19
votes
1answer
595 views

Introducing Tetronogram - Beginner's Version

Introducing Tetronogram! (named by @MrPie) The puzzle is made of a grid like a nonogram. Notations are along the axes like a classic nonogram but numbers are replaced by the names of the ...
8
votes
1answer
224 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 ...
6
votes
1answer
271 views

3D tetromino placement

The following image depicts a $5\times5\times5$ cube. Insert any number of the pictured 3D-tetromino pieces into the cube to satisfy the conditions listed below. Pieces may be rotated in any direction....
13
votes
3answers
846 views

Near-fill with 3x1 long triominos, how to do a different void square than the center square?

It's rather easy to fill a $7 \times 7$ board with 16 long triominos, leaving the center square void: see the picture below. But if I want to move the void square in another position, where else could ...
11
votes
2answers
528 views

BOOM! All Clear for Mr. T

In Tetris 99, Mr. T loves performing All Clears, which happen when a piece clears all lines in the playing field. Being a gentleman, he also tries to minimize the total damage he sends to his ...
11
votes
2answers
308 views

Pentomino solution maximizing straight lines length in rectangle - wood cutter problem

Recently in my free time I cut from wood with my scroll saw two pentomino sets. One set made from 10x6 pattern, and then the other set 20x3 pattern. Think of wood cutter difficulties. I would like to ...
10
votes
1answer
1k views

How many colors does it take?

This question is from a popular monthly science magazine in my country: You have an 8x8 square where any 3 squares forming a tromino (including reflections and rotations) must consist of three ...
5
votes
2answers
797 views

The Pentomino Snake

The premise of the puzzle is quite simple. Here's how to set it up. Draw a 5x5 grid of squares. Write the number 1 in the middle. Make a "snake" of numbers up to 25 so that each number is ...
-5
votes
1answer
215 views

Statue Park (Loop)

Place each of the twelve pentominoes into the grid once, with rotations and reflections allowed. No two pentominoes can overlap or be orthogonally adjacent, and all cells not occupied by the ...
3
votes
2answers
162 views

Tiling rectangles with Hexomino plus rectangle #3

Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ...
14
votes
2answers
553 views

Tiling rectangles with W 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 ...
2
votes
2answers
234 views

Tiling rectangles with a Heptomino plus 2x2 square

Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ...
1
vote
2answers
165 views

Tiling rectangles with Heptomino plus rectangle #7

Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ...
5
votes
2answers
249 views

Tiling rectangles with Heptomino plus rectangle #6

Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ...
11
votes
2answers
508 views

Ziggy - Make a square from 8 polyomino pieces

A few years ago I created a small packing puzzle that I'd like to share here today. The puzzle is based on the fact that $1+2+3+4+5+6+7+8 = 6^2$. It consists of 8 zig-zag polyomino pieces, ranging in ...
3
votes
2answers
146 views

Tiling rectangles with Heptomino plus rectangle #4

Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ...
6
votes
2answers
254 views

Tiling rectangles with V 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 ...
5
votes
1answer
264 views

Tiling rectangles with Hexomino plus rectangle #1

Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 Next ...
2
votes
1answer
139 views

Tiling rectangles with Heptomino plus rectangle #3

Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 Next ...
2
votes
2answers
99 views

Tiling rectangles with Hexomino plus rectangle #2

Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 ...
25
votes
5answers
1k views

Polyomino T hexomino and rectangle packing into rectangle

Let's pack some (one or more) T hexominoes together with some (one or more) small $a\times b$ rectangles into some bigger $m\times n$ rectangle without holes and overlapping pieces. For example, I ...
7
votes
6answers
519 views

Tiling rectangles with N 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 T pentomino plus rectangles ...
17
votes
3answers
315 views

Pairs of Pairs of Pentominoes

Split the 12 pentominoes into three sets of four. Can you pair up pentominoes so that each set makes two of the same shape? For instance, one of your three sets could look like this: That uses the L,...
6
votes
1answer
185 views

Tiling a rectangle with an odd number of Y pentomoes

Follow-on from Tiling a rectangle with just the Y pentomino Two questions: Find the smallest rectangle that can be tiled with an odd number of Y pentominoes, or prove it impossible Find the smallest ...
9
votes
3answers
606 views

Tiling a rectangle with just the Y pentomino

Inspired by this question series, which was inspired by this question. They give rise to beautiful pictures (at least in the eye of the beholder mathematician) and some nice generalizable solutions. ...
15
votes
1answer
432 views

Covering an 8×8 grid with trominoes

I tried to find non-trivial solutions on a deficient 8×8 grid covered with trominoes and I regret to say that after extensive efforts I found only two non-trivial solutions: The conditions of ...
3
votes
3answers
207 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 ...
8
votes
2answers
249 views

Tiling rectangles with T 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 ...
8
votes
5answers
1k views

Polyomino Z pentomino and rectangle packing into rectangle

See my similar question about T hexomino (Polyomino T hexomino and rectangle packing into rectangle) This is exactly same but with other polyomino - Z pentomino. Let's pack some (one or more) Z ...
6
votes
3answers
317 views

Tiling rectangles with X 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 ...
6
votes
2answers
162 views

Four Pieces Polyomino

Goal: Create a symmetric polyomino using: Two pieces Three pieces All four pieces Notes: This is the only set of four different pentominoes which has only one solution with 2, 3 and 4 pieces. ...