Questions tagged [dissection]

A geometric puzzle where a given figure has to be cut into a number of pieces subject to a number of constraints.

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96
votes
2answers
9k views

Prove that π > 3

It seems that once upon a time some politicians tried to pass a law fixing the value of π to be exactly 3. The idea being to "make math simpler so that our children can get better at math". ...
76
votes
5answers
6k views

Can you fold a square into a square of one-fifth the area?

I love origami, and it recently gave me an idea for a very hard but beautiful puzzle. I'm really curious whether anyone here can solve it. So here's the puzzle. You are given a large perfectly square ...
41
votes
2answers
1k views

Dissecting Africa

A straightforward puzzle for the patient. There are no tricks or decryptions needed. The task is 'simple' albeit potentially challenging (and maybe time-consuming). The goal Dissect the Africa-...
36
votes
2answers
4k views

Fairly Sharing a Frosted Cake

You are serving a cake to $10$ children. The the cake is shaped like a box, whose top face is square. The top and sides are covered with a thin layer of frosting. $\qquad\qquad\qquad\qquad$ Every ...
35
votes
1answer
3k views

Cut the disk with a hole in four equal pieces

Cut the shape below in four congruent pieces. The gray area is a hole. In non-mathematical terms, cut the white area in 4 pieces having the same shape, same size, possibly mirrored or rotated. Note:...
34
votes
1answer
3k views

Dissecting the exotic bulbfish

Can you cut the following black shape into exactly three pieces, and then rearrange those pieces into a square?
30
votes
5answers
4k views

Dissection Puzzle - The Umbrella Stand

You own a square-shaped table. You want to drill a small hole in the center to place an umbrella stand. Unfortunately, you're a little drunk: Alas. Fortunately, not all is lost. You are sober now, ...
28
votes
5answers
837 views

Dissecting the holey octomino into a square

This is a pure dissection problem, with no added twists. Cut the holey octomino (i.e., a square with the middle third removed) into several pieces, and reassemble those pieces into a square with no ...
27
votes
4answers
4k views

Drink a Little Wine, Cut a Little Rug

Consider the following diagram of your prized 9' x 12' Persian rug:                       ...
26
votes
5answers
2k views

Pythagorean quilts

The King requests Pythagoras to his palace to discuss an important matter. After the usual formal greetings the King asks: - I have been told that you have a marvelous formula about adding squares ...
26
votes
1answer
1k views

Careless smokers

Cosmo complains: "At the party yesterday at our place, some guys were smoking in our living room. This morning we detected that these careless smokers had burnt four holes into the carpet." Fredo: "...
26
votes
1answer
578 views

You find a piece of paper in your bag

In your bag you find a piece of paper with a size of 5 x 5. You want to make it a 6 x 4 but you may only make 1 continuous cut and reposition the pieces. You are not allowed to bend or twist the ...
23
votes
6answers
4k views

Fair share of a square watermelon?

One hot day, Stan, Kyle, and Kenny were sitting outside with a square watermelon (actually it was a cube like the picture below). Stan says "Let's cut the watermelon into 3 equal slices (like the ...
20
votes
4answers
4k views

Mutilated chessboard

Remove the square in the top-left corner of a $2015 \times2015$ chessboard. Can the remaining mutilated chessboard be tiled with $1\times4$ and $4\times1$ rectangles?
19
votes
1answer
895 views

Restore the square

Triangle ABE is cut from square ABCD (centre E) and placed at one side. Restore the square by a single action different from the exact reversal of this transformation. A solution is to cut triangle ...
19
votes
2answers
715 views

Piece of Cake for King Solomon

Today (7th of July, 2018) marks 40 years of independence for the Solomon Islands. To celebrate, I have decided to bake a cake! And what a pretty cake it's going to be: When viewed from the top, ...
19
votes
1answer
723 views

Dissect the frog

Cut the stylized frog in the picture into six pieces having the same shape and size, possibly mirrored. The white dots are guide points. They help recognizing the shape's geometry. You are not ...
18
votes
3answers
2k views

Tiling a rectangle with nine squares

A rectangle is tiled by nine squares with side lengths $2,5,7,9,16,25,28,33$ and $36$ (without overlapping and without gaps). What are the side lengths of the rectangle? What does the tiling look ...
18
votes
1answer
2k views

Squaring a cross

In this cross shape, all twelve sides are the same length and all angles are right angles. How many cuts does it take to divide the cross into pieces that can be rearranged to form each of the ...
17
votes
5answers
1k views

Universal dissection

Alice has a squared paper 8 by 8. She cuts out one 1x1 square from it, at row N, column M. Bob cuts the rest of the paper into pieces. Once he is done, Alice asks Bob to put the pieces together in a ...
17
votes
3answers
3k views

Can I Haz My Eye Center'd?

Alice has painted a cat on a white disk such that the left eye of the cat is exactly on the center of the disk. Inspired by it, Bob grabbed a white disk of the same size and sat down to paint on it a ...
17
votes
1answer
375 views

Sharing a field among 4 sons

A wealthy famer has a large estate in the shape of an irregular squarish octogon. In the middle he has a rectangular retention basin for storing water. He is getting old and discusses with his wife ...
16
votes
1answer
1k views

The Challenge Square

Q: Can you divide this shape into 4 equal parts, and then form a square?
15
votes
8answers
4k views

Holes in the Table

I was asked this question in an interview and though I was able to answer a part or few parts I'm not really sure if what I gave was the optimal answer. So here goes the puzzle: A restaurant owner ...
15
votes
3answers
898 views

Four fanatics and one checkerboard

Four checkers-playing fanatics eagerly pair up for two simultaneous games of checkers but somehow find themselves with just one board. They do have enough checkers for two games, so it is time to act. ...
14
votes
4answers
3k views

Cutting a 10-by-2 rectangle

How does one dissect a $10\times2$ rectangle into four pieces that can be reassembled to form a square?
14
votes
4answers
2k views

The Erasmus pentagon

Professor Erasmus has constructed a special convex pentagon $ABCDE$ that he modestly calls the "Professor-Erasmus-pentagon". The professor claims that he can cut off a smaller pentagon similar to ...
14
votes
2answers
7k views

Placing 2x1 dominoes on a chessboard with two corners removed

Suppose you have a checkerboard with two opposite corner squares removed, like this: Is it possible to place 31 dominoes of size 2x1 so as to cover all of these squares?
13
votes
2answers
962 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 ...
13
votes
3answers
861 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 ...
13
votes
4answers
1k views

Sharing a Cake with 7, 8, or 9

There may be 7, 8, or 9 guests at a party. The guests will share a round cake (shaped like a cylinder). Define a cut to be any plane or half plane that intersects the cake, i.e. straight cuts only. ...
13
votes
1answer
831 views

Ernie and the Superconducting Boxes

I was in anticipation all last week. Ernie, who had been travelling for several months, was finally coming home. So over the weekend I dropped in on him. He was bursting with news. "Some wonderful ...
13
votes
4answers
1k views

Checker-board Problems

Question 1: Here is a $4\times4$ black and white checker-board. You may fold the checker-board in any direction and times. You may cut the checker-board along a straight line once. The cut needs to ...
13
votes
1answer
614 views

Ernie and the Case of the Singing Sisters

On my way to work each morning I pass a news-agency that usually has a sandwich-board displaying the latest headlines of a somewhat disreputable tabloid newspaper (not that I would ever buy such a ...
12
votes
2answers
2k views

Cut this shape into 3 pieces and fit them together to form a square

A shape is drawn on a sheet of squared paper as shown in the picture below. The shape is then cut from the sheet and given to you. You are asked to first make a straight cut across the shape and then ...
12
votes
2answers
662 views

Polyominoes on a checkerboard

Professor Halfbrain has spent his entire weekend by cutting lots of wooden $50\times50$ checkerboards into lots of polyominoes. He looked at various pattern polyominoes with area $49$, and always ...
12
votes
1answer
636 views

Professor Halfbrain's second cutting theorem

Professor Halfbrain has recently made several fascinating discoveries on cutting convex polygons in the plane. Halfbrain's second cutting theorem: Every convex polygon can be cut (by a perfectly ...
12
votes
2answers
318 views

Unlucky tiling: Arrange thirteen right isosceles triangles into a square

Link to next puzzle in this series:Five graded difficulty isosceles right triangle into square tilings Two difficult "Seventeen right isosceles triangles into a square" tilings V.hard ...
11
votes
2answers
10k views

Divide a rectangle with a rectangular hole into two equal parts

Consider any large rectangle from which a smaller rectangular portion has been removed. The removed rectangular portion may have any orientation i.e. the remaining figure is not necessarily symmetric ...
11
votes
1answer
816 views

Professor Halfbrain and the dissection of a rectangle

Professor Halfbrain has spent his entire weekend with cutting rectangles into smaller rectangles. In particular, he proved the following deep theorem on such dissections. Professor Halfbrain's ...
11
votes
2answers
952 views

Dissecting a square

You are asked to dissect an $N \times N$ square into polyomino pieces such that each piece shares portion of its boundary with exactly $D$ other pieces, and no piece has area exceeding $N$. This can ...
11
votes
1answer
792 views

Professor Halfbrain's first cutting theorem

Professor Halfbrain has recently made several fascinating discoveries on cutting convex polygons in the plane. Halfbrain's first cutting theorem: Every convex polygon can be cut (by a perfectly ...
11
votes
1answer
432 views

Dissect the pixel-heads

Here are two figures, each composed of 54 pixels. Cut each figure into six nonominoes having the same shape and size. Both figures use the same nonomino The nonominoes could be rotated, reflected, ...
10
votes
4answers
1k views

Cutting a square into seven rectangles

Cosmo has cut a square into seven rectangles, so that the seven lengths $\ell_1,\ldots,\ell_7$ and the seven widths $w_1,\ldots,w_7$ of these rectangles satisfy $$ \{\ell_1,\ldots,\ell_7\}\cup\{w_1,\...
10
votes
3answers
768 views

The Erasmus isosceles triangle

Professor Erasmus has constructed a special isosceles triangle that he modestly calls the "Professor-Erasmus-triangle". The professor claims that he can cut his triangle into three smaller triangles, ...
10
votes
1answer
1k views

Dissection of the number-grid (addition/subtraction)

This is the first - presumably simplest - puzzle of an intended series of numerical dissection puzzles. The goal is to dissect the 15 x 11 number-grid below into exactly 17 rectangular sub-grids. ...
10
votes
3answers
403 views

What is the largest number of cubes that can be cut?

Consider a cube made up of 27 unit cubes. If you consider a plane going through the middle of the larger cube it cuts through a number of the unit cubes. The number of cubes that are cut depends on ...
10
votes
1answer
168 views

Combinatorial Agriculture, part 2

Your 64 acre square farm has been doing well and you decide to expand with an orchard. The orchard is also in the form of an 8-by-8 square of subplots, each of which can hold one tree. You have 3 each ...
10
votes
1answer
1k views

Ernie and the Geometric Ginger Cookies

Ernie loves cooking and it's always worth the wait as he follows his curiously precise recipe to make Ginger Crunch Cookies. So I was happy to drop around when he invited me to help test his latest ...
9
votes
2answers
4k views

Dissect a square-and-a-half into 4 equal pieces

The following shape is has the proportions of a square attached to a similar square divided diagonally - A square and a half, if you may. The puzzle is to dissect the shape into 4 congruent pieces. ...