Real Mathematics – Geometry #16

Smell of a Cake

I smell something wonderful. I beam myself up to the kitchen to investigate the source of this smell. I find it: My mother’s chocolate-chipped cake. As I leaned towards the cake, someone grabs my arm: Mom caught me…

I use the emotional card. She doesn’t buy it anymore. My opponent is experienced; my opponent is winning the battle!

As I was thinking of giving up, she offers me a deal. If I can cut three equal pieces out of this cake, one of the pieces will be mine.

Mom’s conditions:

  • Only instrument of measurement allowed for the cut is a compass.
  • Goal is to cut three pieces that have the same area. Size of the pieces is up to my cutting skills.
  • While making the cut, small differences (as if one area is 3,04 and other is 3,09) will be ignored by the mother.
  • Most important condition: Pieces must be in the shape of a ring.
  • You only have one chance for cutting. There is no turning back after the knife touches the cake.

Art of Cutting Cakes

I tried to find a method on paper because I satisfy all the conditions for the cut.

First I drew a circle that has center O:

20190129_223507

Then I created a chord which is as long as the radius of the circle:

20190129_223524

I placed the chord on random places inside the circle and marked chord’s midpoints:

I chose any of those marked points and drew a new circle that has center O:

20190129_223719

Then I followed the same procedure inside the new circle:

And finally I did the same things for the third time:

Areas which I colored with pens are equal to each other:

20190129_224139
Radius of the biggest circle=5 cm.
Radius of the second circle=4,34 cm.
Radius of the third circle=3,56 cm.
Radius of the forth (smallest) circle=2,55 cm.

For those who wonder the areas, you could calculate and see the approximate results.

One wonders…

I found out that it is possible to cut equal areas that are ring-shaped with using only a compass as mother asks.

Now think: How long the chord should be in order to cut the biggest possible piece?

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #8

Right way to cut a round cake

Almost everyone thinks of the same shape when someone mentions “a slice of round cake”:

However, this is not the correct way to cut a round cake. A letter was published in Nature magazine in December 20, 1906. It was written by a famous British scientist Francis Galton. Galton claimed that traditional way of cutting a round cake was faulty as after the cut exposed surfaces of the cake would start becoming dry almost instantly.

keks
Francis Galton’s letter.

Therefore he claimed that he found a “scientific principle” to cut a round cake.

Scientific way of cutting a cake is shown below:

IMG_5755

First Blood: Imagine two lines that are both parallel to the diameter and only a short distance away from it. As Galton suggested, one should cut the cake through these imaginary lines. Hence, exposed surfaces are same and they could be brought together. In order to keep the cake at a stable one piece position, you could use a rubber band.

Second Cut: You can do that like the first cut, but perpendicular to those cuts. In the end you would end up with four pieces of cake. They can be stuck together with a rubber band again.

This is the “scientific” way to keep your cake fresh.

In case you’d like to try Galton’s method without using a cake, all you need is a circle drawn on a paper:

One wonders…

Assume that you are on a Sunday brunch with your friends and pancakes arrived to the table. You realized that the waiter brought one extra pancake and everyone in the table wants that freebee.

24COOKING-CLASSICPANCAKES-videoSixteenByNineJumbo1600

You all agreed to play a game in order to decide who gets the pancake.

Game: Everyone has a pancake on their plate. Everyone will try to cut his/her pancake to most pieces with three straight cuts. (You are not allowed to move the pieces of the pancake.) Winner(s) will get the pancake.

M. Serkan Kalaycıoğlu