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Making a Hard Square Sum Problem Easier

Date: 01/30/2004 at 00:57:40
From: Danny
Subject: Fraction Squares and Triangles

My name is Danny and I am in 6th Grade.  We are working on 
fractions and I have been given the following problem to solve.
Given these 8 numbers:  1/4, 1/2, 3/4, 1, 1 1/4, 1 1/2, 1 3/4, and 2,
I am to place 3 numbers along each side of a square so that the sum of
the 3 numbers on each side of the square is equal to 3.  

It seems like I have tried every combination but when I get to the
fourth side the 3 remaining numbers don't add up to 3.  Is there a
trick to doing this kind of problem?  I really would appreciate some
help.  Thanks!

Date: 01/30/2004 at 11:28:54
From: Doctor Peterson
Subject: Re: Fraction Squares and Triangles

Hi, Danny.

I suspect you are supposed to just do a lot of trial and error, giving
you lots of practice adding fractions.  But there are some tricks that
I used to solve it--mathematicians are always looking for easier ways
to solve problems!

One trick is to avoid fractions entirely.  Notice that all your 
fractions are multiples of 1/4.  What would happen if we had a 
solution to your problem, and then multiplied every number by 4?  It 
would still have the sum of each side the same, but those sums would 
be 4 times as much.  So we can solve your problem by first solving an 
easier problem:

  Arrange the numbers 1,2,3,4,5,6,7,8 around a square so that the
  sum of the three numbers on any side is 12.

    O + O + O = 12
    +       +
    O       O
    +       +
    O + O + O = 12
    =       =
    12     12

That will take a lot less arithmetic to figure out; but there's still 
another trick to use.  Think about what happens when you add all four 
sides together: the sum will be 4 times 12, or 48; but you will have 
added the corners in twice and the side numbers only once.  That is,

  sum of all eight numbers + sum of corners = 48

Do you see that?  From this you can figure out what the sum of the 
corner numbers has to be, and there are only a few ways that can
happen.  So you have only a few possibilities to try.

Once you solve that, you can change your solution into a solution of 
the original problem.  Think about what we did to change it from the
fractions to the whole numbers, and reverse that process to go from
the whole numbers back to the fractions.

This is an example of how mathematical thinking can change a hard 
problem into an easier one.  And that's what makes math a fun 
challenge--it's no fun to do a lot of arithmetic, but it's fun to 
find sneaky ways to avoid it!

If you need more help, please write back and show me how far you got.

- Doctor Peterson, The Math Forum 
Associated Topics:
Elementary Puzzles
Middle School Puzzles

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