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/