# Cutting the cake

Gold Member
I work with a bunch of software geeks. It was someone's brithday last week and we got him a cake, one of those white rectangular ones.

I said he had to cut it into a prime number of pieces, all the same size and shape. How many cuts did he have to make?

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I said he had to cut it into a prime number of pieces, all the same size and shape. How many cuts did he have to make?
I vote for 1 cut!

DaveE

LAF
No cuts (zero) and one piece!
2 cuts and 3 pieces!

Gold Member
No cuts (zero) and one piece!
1 is not a prime number.
2 cuts and 3 pieces!
How will you cut a rectangular cake into 3 same-size same-shape pieces?

Out of a possible 1 points, your score is now -1.

loseyourname
Staff Emeritus
Gold Member
How will you cut a rectangular cake into 3 same-size same-shape pieces?
Say the cake is 3 ft x 1 ft. Make two cuts, perpendicular to the long axis of the cake, at the 1 ft and 2 ft marks as measured from either end of the long axis. You have 3 1 ft x 1 ft pieces of cake.

Gold Member
Say the cake is 3 ft x 1 ft. Make two cuts, perpendicular to the long axis of the cake, at the 1 ft and 2 ft marks as measured from either end of the long axis. You have 3 1 ft x 1 ft pieces of cake.
Good one.

What's the answer if it's chocolate cake?

If it is a chocolate cake I will not bother to cut in such a fashion. Just have whole of it. This question has infinite answers.

Gold Member
If it is a chocolate cake I will not bother to cut in such a fashion. Just have whole of it. This question has infinite answers.
Not really. The question asks how many he had to cut. The implication is that the answer assumes no more cuts than necessary to meet the conditions.

Also, some answers are going to be trvially similar.They can be generalized using algebra to result in a small and very finite number of answers.

For a white, rectangular cake:
a single cut which passes through the center of the cake will cut it into 2 pieces. Since 2 is a prime, this should suffice. The interesting thing is that it doesn't matter what angle the cut makes with the cake, the two pieces will be the same size and shape.
Chocolate cakes work the same.

I like Serena
Homework Helper
If the number of guests is the prime number p, you are forced to make p - 1 cuts that are parallel to each other.
The only exception is p = 2, where any cut through the center is ok.
[EDIT]
Note that the same holds for a circular cake where you would cut from the center outwards.
In this case the cuts do not need to be straight as long as rotational symmetry is observed.
In the rectangular case, all cuts need to be straight, because we need to observe translational symmetry and the sides of the cake are straight.
The exception is for p = 2, where any cut through the center works as long as it is point-symmetric.
[/EDIT]
-- I like ILSe

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