# Questions tagged [packing]

A puzzle that involves arranging objects optimally in order to fit in a specified space.

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### Packing 25 three-dimensional N pentominoes into a 5x5x5 cube

The puzzle contains 25 identical pieces that look like this: To be explicit, the piece is composed of five cubes. In the picture, three cubes form the base, and two cubes form the overhang. The goal ...
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### Build a slanted pyramid with ten L-shaped blocks

Consider the following L-shaped 3-dimensional object made up of three unit cubes joined at their faces: Use 10 of the above L-shaped pieces to make the following shape:
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### Can 42 1x2x4 cuboids be packed into a 7x7x7 cube?

Can 42 1x2x4 cuboids be packed into a 7x7x7 cube without cutting any of them? Assume that all cuboids have their axes parallel to the axes of the big cube. I tried using https://www.jaapsch.net/...
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### 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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### Pawns and a chessboard with no three aligned

This little problem crossed my mind and appeared to be not quite trivial. How can you place P pawns on a chessboard with the constraint that no pawn is exactly midway between two other pawns? Sure ...
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### Packing a 3x3x3 cube with three congruent polycubes

A polycube is a solid three-dimensional connected figure formed by joining one or more unit cubes face to face. Polycubes can be thought of as the three-dimensional generalization of polyominoes. Can ...
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### PSE Advent Calendar 2023 (Day 8): A Quilt for Santa

This puzzle is part of the Puzzling Stack Exchange Advent Calendar 2023. The accepted answer to this question will be awarded a bounty worth 50 reputation.< Previous Door Next Door > Mrs. Claus ...
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### Packing a 3xMxN-box with copies of a single tetracube

There seems to be no solution for packing a 3xMxN-box with copies of this tetracube. Does anybody know of a proof that this is impossible or is this still an open question?
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### 12 piece cube packing puzzle

Consider the following hexacube (made from 6 unit cubes): GOAL: Pack a 3 x 3 x 3 cube using three of these hexacubes plus nine unit cubes. This puzzle comes from: https://puzzlewillbeplayed.com/333/...
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### Can you pack these pentacubes to form a rectangular block with at least one odd side length other the side whose length must be a multiple of 5

This puzzle is part of the Monthly Topic Challenge #11: Now in 3D. Consider the following pentacube (made from 5 unit cubes): It is possible to pack four of these pentacubes to form a 2x2x5 ...
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### How many ways are there to solve the Mensa cube puzzle?

The Mensa cube is a puzzle in which a solid cube has been partitioned into $N=11$ rigid parts. The goal of the puzzle is to re-assemble the cube from its parts and place it back in its rigid box. See ...
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### Can you pack these tetracubes to form a rectangular block with at least one odd side length?

Consider the following tetracube (made from 4 unit cubes): It is easy to pack two of these tetracubes to form a 2x2x2 rectangular block. And from that simple packing it is easy to pack any ...
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### Cutting a square into integer triangles

You are given a square piece of paper with size 10x10 units. What is the most number of triangles that can be cut from this square, such that: Each triangle has integer sides. Each triangle is ...
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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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### Boxeslayers to the rescue

This is about layering boxes, not about slaying them. We have 1,830 2×5 boxes to stack safely as 10 alternating contiguous layer patterns of 183 boxes each. Layers have identical silhouettes that fit ...
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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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### Arranging shapes into a similar shape

The goals if possible. Goal 1. In the image there are 12 shapes each containing 15 cells. Take any 3 shapes from the set and arrange them into any new shape, 2 example shapes that you could use 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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### Smallest square that can pack thin digits

If we draw the digits 0 to 9, segmented into squares, across a rectangle of 2x5 (except the 1) they use up 81 total squares. Is it possible to pack them all into a 9x9 grid. What is the smallest n by ...
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### Smallest rectangle that fits the first 10 rectangles [closed]

What is the area of the smallest rectangle that can fit 10 rectangles with areas 1 to 10, inclusive? Rectangles must have integer sides and cannot overlap.
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### Ernie and the disappointing drill bits

I was driving to Ernie's place a couple of days ago, with Ernie in the passenger seat, when we drove past an old building with a faded sign reading Lar's Tool's. Beneath it, scrawled on the front wall ...
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### Packing the Primes

Here is a 5x4 rectangular wordsearch (area 20) containing the primes between 1 and 100, {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97} (numbers in any ...
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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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### 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 ...
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### 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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Due to some unfortunate events involving a teleporter and a ceiling fan I had my innards spread all over the place. Interestingly my intestine was split into equal sized chunks representing all the ...
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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 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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### 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 ...
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### Put three pieces of cake into a round box

You're about to cut three pieces from a large cake to put in a round box of radius 1. If the pieces must be congruent triangles, and cannot overlap, what shape gives you the maximum amount of cake?
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### Dividing a piece of land

Alice and Bob try to divide a piece of land $D$, shaped in a perfect closed disk of radius 1. Alice moves first to mark some finite (at least one) number of points in $D$. Bob then draws any number of ...
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### Can you stop the falling piano with 23 nets?

MIT's Baker House has a tradition of dropping an irrepairable piano six floors every Drop Day, the last day one can drop a class without penalty (the 2022 date is 19 April). This year, in order to ...
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### How many squares can a limp queen move to?

Consider a large chessboard. A limp rook is a chess piece that moves one step orthogonally, but it turns $90$ degrees after every move. The limp rook makes some moves, not crossing over its own path, ...
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### Jigsaw puzzle: packing pentominoes into a rectangle

I've got this jigsaw puzzle that I can't figure out. The major problem is that there are no signposts on whether a piece is in the right place. How does one get all the pieces into the 6x10 container? ...
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### Social distancing in a 5x5 room [duplicate]

I have booked a square meeting room that is 5 by 5 meters. Our Covid-19 policy says that each person must be at least 1.5 meters away from any other person. What is the highest number of people that ...
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### Arrange the wooden blocks in the box

Little Johnny is the naughtiest child in the kindergarden today. The kind who always wants more than the other children. When he saw that little Tim was playing with the same set of wooden blocks as ...
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### Eighteen is not seventeen

This question is not the same as Adding coins inside a ring of coins From a pile of equal size perfectly round coins take eighteen and make a perfect ring. Show that you can fit at least sixteen more ...
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### Adding coins inside a ring of coins

17 identical coins with diameter 1 are lying flat on a table, such that their midpoints build the vertices of a regular 17-gon (regular heptadecagon) and adjacent coins touch each other. What is the ...
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### How many figures to cover the kitchen floor?

The puzzle is as follows: Figure 1.1 shows a kitchen floor that is made up of 30 squares whose sides measure 1 cm in length and in Figure 1.2 a tile that is made up of 5 squares whose sides also ...
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### What is the name or author of this tangrams-like packing puzzle?

Does someone knows the name of this packing puzzle, or its inventor? The goal is to repack the shapes into the square with the additional red square. It is an easy puzzle but the construction is ...
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### Can a square of size 1000.25 fit a million and one unit squares?

A square with a side length of exactly 1000 can obviously be packed with exactly a million unit squares. If we increase the side length to 1001, then 2001 more squares can fit. But if we increase the ...
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### A packing game!

Amy and Ben are playing a game which is suggested by a genie. Amy first chooses $a,b,c\in\mathbb{R}^+$. Then a empty cuboid box with internal measurements $a+b,b+c,c+a$, and infinite supply of cuboid ...
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