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### Cutting a Square

```
Date: 05/05/99 at 10:50:56
From: Nickolas Joubert
Subject: Maths investigation - "cutting cake"

I have been given this math investigation to do. I am completely stuck
and I wondered if you could help me.

Given a square that is any size, I have to "cut" the square ten times
and see how many pieces I can obtain. I have to obtain as many pieces
as possible. For example, a square can be cut into two with one
straight cut. That same square can be cut into four pieces with two
straight cuts. However, I can obtain seven pieces with three straight
cuts. I hope you see what I mean.

I have worked out a pattern and have found out how many pieces can be
obtained from ten cuts, which is fifty-six. I understand it up to
there, but now I need to work out a formula. I need the formula so
that I don't need to go through a lot of work to out to find out how
many pieces can be obtained from a much higher number of cuts, for
example, seventy.) I would be very grateful if you could help me.

Thanks very much,
Nickolas Joubert
```

```
Date: 06/11/99 at 22:30:00
From: Doctor Jeremiah
Subject: Re: Maths investigation - "cutting cake"

Hi Nickolas:

If your pattern is 2, 4, 7, 11, 16, 22, 29, 37, 46, 56... then we
should be able to find a formula.

I don't know for sure whether this pattern is right or not. It does
look like it though. I am sure that 2, 4, 7, 11, 16 is right (I
checked them myself).

Lets assume that the pattern holds for bigger numbers.

To create a formula for this we need to figure out what we are doing
to one number to get the next number. This is easy; we are always
adding a value one bigger than the last number we added.

2 = 1 + (1)
4 = 1 + (1 + 2)
7 = 1 + (1 + 2 + 3)
11 = 1 + (1 + 2 + 3 + 4)
16 = 1 + (1 + 2 + 3 + 4 + 5)
22 = 1 + (1 + 2 + 3 + 4 + 5 + 6)
29 = 1 + (1 + 2 + 3 + 4 + 5 + 6 + 7)
37 = 1 + (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ... is a summation. The formula for
this summation just happens to be n(n+1)/2, where n is the largest
number you are adding.

2 = 1 + n(n+1)/2  <==  n = 1
4 = 1 + n(n+1)/2  <==  n = 2
7 = 1 + n(n+1)/2  <==  n = 3
11 = 1 + n(n+1)/2  <==  n = 4
16 = 1 + n(n+1)/2  <==  n = 5
22 = 1 + n(n+1)/2  <==  n = 6
29 = 1 + n(n+1)/2  <==  n = 7
37 = 1 + n(n+1)/2  <==  n = 8

So n is the number of cuts, and 70 cuts would make:

1 + 70(70+1)/2
= 1 + 2485
= 2486 pieces

If you want to know where the n(n+1)/2 comes from let me know.

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

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