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The Holey Square Family

Put-Together Puzzles

by Serhiy Grabarchuk

One of the simplest geometric paradoxes with vanishing areas was described by Martin Gardner in his classic book, Mathematics, Magic and Mystery, (1956).1 The paradox is more than two centuries old, but it still attracts attention of puzzlers, mathematicians, and magicians. And that is why there are dozens of different variations based on its principle.
Important: All puzzles are published here with the kind permission of their authors, where applicable. Copyright to all presented puzzles stays with their respective authors, unless otherwise is stated.

Printable Pieces    

Printable Puzzle: To solve some of the puzzles which will be described below you can print their pieces. For this, click the respective image marked with the "pp" pictogram (as that shown at left) to go to a new window with the puzzle tiles; then you can print them and cut them out.

Copyright Note

Copyright Note: Note that all presented puzzles are copyrighted, so you can print them for your own use only, and not for any kind of commercial profit.

Play Puzzle    

Play Puzzle: Click an image marked with the "pp" pictogram (as that shown at left) to go to a page with an interactive version of the respective puzzle.
The Holey Square
The basic idea is that a square is dissected into four pieces so that they can be reassembled into another square with seemingly the same sizes, but with a square hole in it.
         Printable Pieces
The Holey Square without and with a hole in it. The sample was hand-crafted by the author in the early 70's.
There are many versions of this puzzle-paradox based on the described principle, but with improved pieces. Some versions use a rectangle instead of a square; among them several versions exploit a playing card theme, while some other use an internal rectangular piece with some pictures, inscriptions, or even as... a business card(!). Also, there are clever, 3D versions of this classic puzzle-paradox. Using an asymmetric pattern for dividing a plain square, we can make amazingly tricky puzzle-paradoxes. The simplest of them is described just below.
The Coin Puzzle-Paradox

   by Serhiy Grabarchuk

Four quadrilateral pieces with coins pasted on both of their sides make a square with a hole in the middle. One more coin is placed within the hole. If you will take that coin away, the four remaining ones make a square with a free square hole within. Now, you can rearrange the four pieces so that big square appears again. And again four coins make the same square of the same size and in the same orientation as in the initial pattern, but now no hole remains anymore. How can that be possible?
Printable Pieces
The square with five coins and a hole, and the new square with four coins and no hole in it.
You can prepare the pieces to perform the trick. For this click the above image to open the unfolded patterns of the pieces, print them, cut them out, fold four of them along the dotted lines in half, and use some glue to join the halves in each piece together.
Play Puzzle

Where Is the 5?
Where Is the 5?2

   by Serhiy Grabarchuk

It is a simple trick with four quadrilaterals. They make up a square with a traditional magic square, 3 x 3 on it; its magic constant is 15, what means that each of its rows, columns, and both main diagonals add up to 15. When you shuffle the pieces, and then flip them over, you will see fragments of another, 3 x 3 magic square. Restore it and you will see how one of the numbers disappears... To learn more click the thumbnail just at left.

A Holey Number.
A Holey Number3

   by Werner Miller

This puzzle-paradox is just similar to the above described trick, and combines a holey square and a 4 x 4 pandiagonal magic square. The four quadrilateral pieces with printed numbers 0 through 15 are assembled into a square with a 4 x 4 magic square on it; each of its rows, columns, and both main diagonals add up to 30. Then pieces are flipped over, and again they can produce a magic square with the same magic constant 30, although now the whole square 4 x 4 lacks one cell. Is not that odd?... Click the thumbnail just at left to learn more.
Puzzle-Paradox Mosaic

   by Serhiy Grabarchuk

This Tangram-like set is based on the Holey Square's pattern with each of its quadrilaterals divided into two different right triangles--an isosceles triangle, and a triangle in the size of a half-domino. Note that the area of the full square is 10, and so its side is sqrt10. Using the full set every time, we can form two different squares (with sides 1 and 3); two equal squares (with sides sqrt5); different sets of three and four squares; and a lot of different shapes. You can flip pieces over, but not overlap.
Printable Pieces
The Puzzle-Paradox Mosaic consists of nine pieces: eight right triangles of two different kinds, and a single square.

Assemble these shapes, using all the nine pieces of the Puzzle-Paradox Mosaic.
Notes & References
     1) Martin Gardner, Mathematics, Magic and Mystery, Dover Publications, Inc., New York, 1956, (see page 151).
     2) First this puzzle as an
interactive Flash version and a printable PDF was published at Puzzles.COM.
     3) This version was created by Werner Miller, and published at
Visions website, The Online Journal of the Art of Magic, as an article in its department, "In Your Hands."
Some specific variations of the Holey Square puzzle and their properties are discussed in Stewart Coffin's article, "Polly's Flagstones," published in
The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner, A K Peters, Ltd., Natick, Massachusetts, 1999, pages 103-105.
Henry Perigal's dissection of two squares to one, and some other dissections which use the Holey Square's principle are described in Greg Frederickson's famous book, Dissections: Plane and Fancy, Cambridge University Press, 1997, Chapter 4: It's Hip to Be a Square, pages 28-39.
     In 2010 Puzzle-Paradox Mosaic was nicely
origamized by Francis M. Y. Ow.
Last Updated: February 7, 2014
Posted: October 14, 2008
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