Chocolate cutting trick

https://www.youtube.com/watch?v=dmBsPgPu0Wc

I recently watched this video which showed someone cutting up a bar of chocolate. Originally I questioned where the maths was in this, until I looked further into this. I watched this video several times and tried to work out how a bar of chocolate could be cut up, with a piece taken away but still fit together looking as if the bar of chocolate never changed at all (I wasn’t complaining that bar didn’t get smaller!)

I wrote a list of instructions to work this puzzle out:

1. Remove the red rectangle
2. Move the green trapezoid to the left
3. Move the blue trapezoid to the right

This is the bar of chocolate which is not yet cut, but shows the lines where it would be cut into the different pieces. I then moved all the pieces like in the video and discovered that the car of chocolate, unfortunately does get smaller. But how does that work when it still looks the same? The picture below shows that when you take the single rectangle away, the bar of chocolate gets smaller. So how do this work? What’s the mathematics behind it?

So, where’s the maths in all of this?

First, we need to find the area of the original bar of chocolate.

Length = 6cm

Width = 4cm

Area = Length x Width

Area = 6cm x 4cm

Area = 24cm²

So, to calculate how much surface area is left when the piece is taken away we need to work out what size each piece of chocolate is and in this case, I have worked it out to be 1cm. So, finally to work this out is rather simple. All that we need to do now is find the surface area of the single piece which is 1cm x 1cm which is 1cm² and then take this away from the total surface area.

24cm²  – 1cm²  = 23cm²

So the infinite bar of chocolate, isn’t really infinite at all!