How Many Border Stamps?

Date: 04/17/2001 at 17:11:15
From: Emilee
Subject: Multiplication  (about a sheet of stamps)

Dear Dr.Math,

I have a difficult question and I don't have a clue how to do it: 

How many stamps are on the border of a 10x10 sheet of 100 stamps?

Sincerely,
Emilee

Date: 04/19/2001 at 12:31:50
From: Doctor TWE
Subject: Re: Multiplication  (about a sheet of stamps)

Hi Emilee - thanks for writing to Dr. Math.

Here are two ways of thinking about the problem. Maybe one of these 
will click for you.

Method 1: Draw a picture of the problem.

In this case, it's not too hard. Sometimes problems involve such large 
numbers that drawing a picture is not a practical alternative, but 
drawing 100 stamps isn't too bad. Here they are, arranged in a 10x10 
pattern (each [] represents 1 stamp):

     [] [] [] [] [] [] [] [] [] []

     [] [] [] [] [] [] [] [] [] []

     [] [] [] [] [] [] [] [] [] []

     [] [] [] [] [] [] [] [] [] []

     [] [] [] [] [] [] [] [] [] []

     [] [] [] [] [] [] [] [] [] []

     [] [] [] [] [] [] [] [] [] []

     [] [] [] [] [] [] [] [] [] []

     [] [] [] [] [] [] [] [] [] []

     [] [] [] [] [] [] [] [] [] []

Now all we have to do is count the number that are on the border. You 
might want to mark the stamp you start with, so that you don't 
accidentally count right past it and keep going!


Method 2: Use what we know about squares and rectangles.

We know that a square or rectangle has four sides. In this problem, we 
know that each side (or "border edge") has 10 stamps along it, so we 
have 4 x 10 border stamps... or do we? We have to be careful, because 
each stamp that's in a corner is part of two sides, so we've counted 
those stamps twice. To fix that, we subtract the number of corners, 
which is 4. Then we have:

     4 x 10 - 4

border stamps.


Try each method and see if you get the same answer. Can you think of 
any other ways to get the answer?

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/