The Perfect Ten Paperclip Paradox is a mathematical puzzle that was published in the book, The Art of Astonishment- Volume 3 by Paul Harris. 

The explanation for this puzzle is due to the arrangement of the clips. When counting the clips, the ones in the corner are counted twice, as they are part of a horizontal row and a vertical side. The clips between the corners are counted only once during the counting process. When a clip is added to a row or side, there are initially eleven clips in the row or side. However, when a clip is then moved from a corner to a place between the corners, it is only counted once, which keep the numbers of clips at 10 when counted for each row or side. A video explanation is also available for this puzzle by using the link.

This magic puzzle was presented by Dieter Kadan from Austria, at the 2019 Science on Stage festival. It is based on a magic puzzle invented by a New York Magician named Paul Curry, although the principle of this paradox has been known since the 16th century (https://en.wikipedia.org/wiki/Missing_square_puzzle).

When the shapes are assembled, the number of squares in the rectangle is 65, the square has 64 squares, and the final shape has sixty three squares. It appears the squares are vanishing each time the pieces are reassembled!

Explanation

The true number of squares is 64 as seen in the square shape. When the shape is assembled into a rectangle, the pieces don’t align completely. There is a small amount of space from the top left to bottom right diagonal. This space is equivalent to the area of one extra square, giving the illusion of 65 squares in this shape. When the other shape is reassembled, there appears to be 63 squares, as the pieces have to overlap each other slightly to form the shape. The area of overlap is the same as one whole square, so it appears there is one square less.

This is a nice way to show how care and precision are required in science to generate accurate data. If care isn’t taken when gathering data, then the results can be variable as shown when trying to count squares in shapes if the pieces aren’t aligned properly.

Try this!
1: Think of a number from 2 to 10
2: Multiply your number by 9
3: Add the two numbers of your total together
4: Take 5 away from your answer
5: If your answer is 1, it equals A; if it’s 2, it equals B; if it’s 3, it equals C; if it’s 4, it equals D (no need to go any higher, as the answer will of course always be D)
6: Think of a country in Europe beginning with your letter
7: Think of an animal, not a bird or a fish, beginning with the second letter of your country
8: Think of the colour of your animal

Most of you will be thinking of a grey elephant in Denmark!

This is a self-working maths trick. Any single digit number multiplied by 9 will give two-digit number. If you add the digits of the two-digit number, it will always add up to 9.

So, if you choose 7, 7 x 9 = 63, 6+3 = 9. If you subtract 5 from 9 you will get 4.

D is the fourth letter of the alphabet. The country most people will think of is Denmark. Don’t give people too much time to think or they may come with Djibouti or Dominican Republic. The second letter of Denmark is E so most will think of elephant when asked for an animal beginning with E. The colour of an elephant is grey which you announce to the class as evidence of your mindreading abilities.

1. Draw the following numbers, on your piece of paper:

2. Find a way to keep your paper standing;

3. Put the glass in front of your numbers;

4. Pour water into your glass until the level of the water is above the numbers;

5. Move the glass towards and away from you until you find the spot where the numbers are reversed;

6.     Once you’ve worked out where that is you are ready to perform the trick.

7. To perform the trick, cover the glass with a small card or handkerchief before adding water to create a greater effect.

This trick can be used in Pre-School and Primary to talk about shapes using a story. 

For older students we can use this trick/illusion to talk about perspective. 

How about using this story for the little ones and adding this illusion?

"Once upon a time there was a square named Claire. Claire liked to play  and explore. One day, something magical happened. Claire started to change. She became a curve, Claire giggled with joy, rolling like never before. She discovered a new way to play. Bute as the sun began to set, Claire wanted to be a square again. With a wish and a twist she turned back."

This trick can be used in Pre-School and Primary to talk about shapes using a story. 

How about using this story for the little ones and adding this illusion?

“Once upon a time, there were two intertwined circles called Twirly and Whirly. They always saw things differently and neither could understand the other's point of view.

One day, they realized that these differences were tearing them apart. They decided to take some time to understand each other and work together.

With patience and some tweaking, counting their differences, they found a way to align their views. In doing so, they magically transformed themselves into a frame and both of them, at the same time, could understand each other because they were looking to the same thing at the same time.

We learn from Twirly and Whirly, we can overcome differences and that we can find a way to see the world in the same way and understand each other's point of view.”

Give a volunteer from the audience an envelope with a prediction.

Show 9 cards numbered from 1 to 9. Ask a member of the public to choose 3 cards. 

Write the chosen numbers on a board or sheet of paper. For example, if the numbers 2, 5 and 8 have been chosen, the rule is to put the largest number on the left (but this instruction should not be communicated).

Write down the number 852.

Now ask them to subtract the inverse, i.e. 852 - 258. Write the result 594 and add the symmetric. 

Write down the result 1089.

Now ask the person you gave the envelope to open it and ask them to read the number written on it: it's 1089.

Let us assume that the initial number is the larger and has digits a, b and c. So, when we reverse and subtract we will have (100a + 10b + c) – (100c + 10b + a)

This is the same as 100a +  10b + c – 100c – 10b – a = 99a – 99c = 99(a – c)

Because a and c are integer numbers, at the end of the first part of the process we will always end up with a multiple of 99.

The three digit multiples of 99 are: 198, 297, 396, 495, 594, 693, 792 and 891.

Now, note that the first and last digits of each number add up to 9.

So, when we reverse any of these numbers and add them together we get 9 hundreds from the first digit, 18 dozens from the second digits and 9 units from the third digit.

So we get 900 + 180 + 9 = 1089.