Focus on...Dudeney’s Greek cross dissection puzzles
“Puzzles have infinite variety, but perhaps there is no class more ancient than dissection, cutting-out, or superposition puzzles. They were certainly known to the Chinese several thousand years before the Christian era.
All good dissection puzzles are really based on geometrical laws – geometry will give us the ‘reason why’. I have known more than one person led on to a study of geometry by the fascination of cutting-out puzzles.
The fact that they have interested and given pleasure to man for untold ages is no doubt due in some measure to the appeal they make to the eye as well as to the brain. There are probably few readers who will examine cuttings of the Greek cross without being in some degree stirred by a sense of beauty.”
Henry Ernest Dudeney, from Amusements in Mathematics, published in 1917.
Dudeney tells us that the Greek cross, with four equal arms and formed from five equal squares, was regarded for thousands of years as “a sign of the dual forces of Nature - the male and female spirit of everything that was everlasting”.
A Greek cross puzzle, known as the Hindu problem, is supposed to be three thousand years old. It appears in the seal of Harvard College. The challenge is:
cut the cross into five pieces that will form a square.
Does each cut have to be from a corner to the mid-point of a side?
Or is this also a solution?
Can you justify what you believe by thinking about half-turns?
The same puzzle with only four pieces was not solved until the middle of the nineteenth century: cut the cross into four pieces that will form a square.
Dudeney wrote in 1917 “Here we have the great Swastika, or sign, of ‘good luck to you’ - the most ancient symbol of the human race of which there is any record. One might almost say there is a curious affinity between the Greek cross and Swastika!”
(This, of course, was before it was misappropriated by Nazi Germany in the 1920s and 30s.)
Must each cut be from the mid-point of one side to the mid-point of another side?
Or are there other solutions, such as this?
In the previous example, you need to cut the card or paper three times. Learners can make harder puzzles by adding further conditions. For example:
cut the cross with only two straight cuts into four pieces that will form a square.
For every challenge to cut the cross into pieces that will form a square, you can ask the reverse question; how do I cut a square into pieces that will form a Greek cross?
For example: cut a square into five pieces that will form two separate Greek crosses of different sizes.
By exploring possibilities on a square grid, a student might arrive at this solution:
A learner might modify the previous challenge, for example by changing it to: cut a square into five pieces that will form two separate Greek crosses of exactly the same size.
A possible strategy is to consider the reverse problem – how to form a square from two identical Greek crosses. A student who sketched a Greek cross on a square grid visualised this:
Mentally splitting the red cross into four identical pieces with perpendicular cuts led to a solution:
This way of ‘seeing’ how to cut a square into five pieces to make two identical crosses may suggest how to: cut a square into four pieces that will form two separate Greek crosses of exactly the same size.
It is natural to visualise the same square grid superimposed over the square.
In Amusements in Mathematics, Dudeney sets out to show readers how the side-length of the square that has the same area as a Greek cross is related to the dimensions of the cross, and gives an explanation. Students may discover this themselves, and then, perhaps in response to a little questioning, explain their thinking in their own ways. When students try to convince each other that what they believe is true, and explain their reasoning, they are truly doing mathematics.
This is Dudeney’s explanation, using Pythagoras’ Theorem:
Imagine moving the bottom square of the cross to complete the square on the green side of the right-angled triangle. The area of the cross is the area of the square on the red side plus the area of the square on the green side – which is the area of the square on the blue side of the right-angled triangle. So the area of a Greek cross is the same as the area of the square on the diagonal of a rectangle composed of two of the five small squares making the cross.
Dudeney also tries to convince readers that there are infinitely many different ways of cutting the cross into four pieces that will form a square. He asks us to visualise, placed over a Greek cross, a grid of squares equal to the square that has the same area as the cross. Providing that you always have the grid lines parallel and perpendicular to the diagonal of a two-square rectangle in the cross, and a point of intersection of the grid within the central small square of the cross, the grid lines will give you the lines of the cuts to make the four pieces. Because there are infinitely many points within the central square of the Greek cross, there are infinitely many ways of cutting the cross into four pieces that fit together to form a square.
Students will see that this always works in every position of the grid, under Dudeney’s conditions, that they try. But do they understand WHY it works!
Dudeney reminds us that Pythagoras’ Theorem also explains the following dissection.
Cut a shape made from two squares of any relative dimensions into three pieces that will fit together to form a square.
Both cuts are from a point – on the longest side of the shape – that is the same distance from the corner of the larger square as the side-length of the smaller square. (The green and blue triangles are congruent.)
Because the start shape can be made from two squares of any relative dimensions, the two squares might be identical, so that the shape is a rectangle. We then have this special case of the previous diagram:
Dudeney writes: “If you make the two squares of exactly the same size, you will see that the diagonal of any square is always the side of a square that is twice the size. All this, which is so simple that anybody can understand it, is very essential to the solving of cutting-out puzzles. It is in fact the key to most of them. And it is all so beautiful that it seems a pity that it should not be familiar to everybody.”
The following three Greek cross dissection puzzle may challenge students. But if they have explored the previous dissections, they may solve these by thinking about their previous arguments, findings and successful strategies!
Take a square and cut it in half diagonally. Now try to discover how to cut this triangle into four pieces that will form a Greek cross.
Cut a rectangle of the shape of a half-square into three pieces that will form a Greek cross.
Cut a Greek cross into five pieces that will form two separate squares, one of which contains half the area of one of the five squares that compose the cross.
Having previously reminded us of what your students may have observed – “That the diagonal of any square is always the side of a square that is twice the size,” Dudeney now prompts us with “It is also clear that half the diagonal of any square is equal to the side of a square of half the area.” Therefore, if the (pale green) square in the diagram below is one of the five squares of the Greek cross, the small square on half its diagonal is the size of one of the squares required in this puzzle!
Dudeney ends his writing (in Amusements in Mathematics) about Greek cross dissections with the following statement:
“I have thus tried to show that some of these puzzles that many people are apt to regard as quite wonderful and bewildering, are really not difficult if only we use a little thought and judgment.”
Then he provides four more Greek cross puzzles!
- 142 The silk patchwork
- 143 Two crosses from one
- 144 The cross and the triangle
- 145 The folded cross
These can be found at Gutenberg.org.