Walk through a piece of paper - The Math behind it

Today, as part of our math lessons, I asked my kids, “Do you think I can cut a hole large enough in a piece of paper such that I can walk through it ?” They immediately replied “No way, you can’t “. Hmm, I told them, let’s do some Math Trick today. ‘Walk through a piece of paper’ is one of the math tricks that makes you wonder how it works. It is basically based on the mathematical concept of topology which involves looking at the properties of geometric objects that remain unchanged after deformation like shrinking, stretching and folding but not tearing. In this math trick, we’re going to look at the area and perimeter properties of a rectangle.

Materials

A4 sheet paper
Scissors

You can also print out a template to make it easier for young children HERE. The aim is to cut the paper in any shape you think suits, but you can’t cut the paper all the way through it. You can not use any other material and you can’t glue or fix any of the paper.

Instructions

Fold the paper in half along its long side. Place the paper with the folded side closet to you.

Starting about 1 cm from one end of the paper, cut a slit from the folded side of the paper leaving 1 cm uncut at the end so you don’t cut all the way across the paper.

Turn the paper around so the fold is away from you. Along the loose end, cut another slit 1 cm apart from the last cut, and stop about 1 cm before you get to the folded side.

Alternate between cutting from the folded end and the loose end and finishing on the folded end until you’ve reached the other side of the paper. Keep all the alternating cuts 1 cm apart and make sure you don’t cut all the way through.

Now, you will notice a series of folded slits along the folded side, cut along the fold of each of the slits except the first slit and the last slit. Leave them untouched.

Carefully unfold the paper and you should have a large loop that you can fit your body through.

The Math Behind It

By cutting a long continuous maze like path in the paper, you have created
a “hole” that is apparently larger than the paper itself. We know that multiple rectangles can have the same area, yet their perimeter will be different. For example, a rectangle of 10 x 20 cm has an area of 200 cm and a perimeter of 60 cm. Similarly, a rectangle of 1 x 200 cm has the same area of 200 cm but a perimeter of 402 cm. Using the same concept, you have created a loop that is big enough for you to walk through. The area of the sheet of paper remained the same throughout the activity – the size of the paper was not changed at all because none of the cuts went all the way across the paper

The original paper perimeter did not allow you to cut a hole that would be large enough to fit through. However, a new and much longer perimeter is created by cutting slits back and forth in the folded paper. The length of this perimeter is limited to how narrow a slit can be cut. The narrower the slits, the longer the perimeter, and the bigger the hole.