Solving Number Sentences: A Strategy

Date: 07/11/2003 at 17:53:10
From: Melissa
Subject: How do we figure number sentences

I have been given a list of numbers without any plus or minus signs 
and I have to place the signs between the digits. Here are a few 
problems:

1  9  8  6  3  5  1  =  6
3  5  3  2  4  1  5  =  2
5  5  1  1  3  4  8  =  18

I can't figure out which sign goes where. 

Date: 07/11/2003 at 23:04:28
From: Doctor Peterson
Subject: Re: How do we figure number sentences

Hi, Melissa.

This may be meant as a way to make you practice adding and subtracting 
over and over until you find the answer. But if you step back and 
think about how numbers work, there is a nice little trick that can 
save a lot of time.

Let's take a slightly smaller example. Suppose we want to combine 
these numbers to make 4:

  4 _ 2 _ 3 _ 7 _ 6 _ 4 = 4

Think about how we add and subtract. If we calculated this, for 
example,

  4 + 2 - 3 + 7 - 6 + 4 = 8

we would be adding 4, 2, 7, and 4, and subtracting 3 and 6. It turns 
out that it doesn't matter what order you do that in; I actually 
calculated it as

  4 + 2 + 7 + 4 - 3 - 6 = 8

doing all the subtractions last. And in fact, subtracting 3 and then 
6 is the same as just adding 3+6 and subtracting that:

  (4 + 2 + 7 + 4) - (3 + 6) = 8

So any combination of addition and subtraction really amounts to 
subtracting the sum of some of the numbers from the sum of the others.

Now, let's look at our numbers. The sum of ALL of them is

  4 + 2 + 3 + 7 + 6 + 4 = 26

So we want to find a pair of numbers whose sum is 26 and whose 
difference is our goal, 4. There are several ways to do that; I'll 
let you think about how you might do it. (I have a great trick for 
that, too!) But the answer is that 15 + 11 = 26 and 15 - 11 = 4. So 
all we have to do is to find a group of numbers from this set that 
add up to 15, and then subtract the rest of them.

I see that 4 + 2 + 3 + 6 = 15; or, working from the middle, 7 + 4 + 4 
= 15. So that gives me two answers (and there might be more):

  4 + 2 + 3 - 7 + 6 - 4 = 4

  4 - 2 - 3 + 7 - 6 + 4 = 4

See if you can solve your puzzles a little more easily this way. It 
does take a lot of thinking to invent the trick, or to understand it, 
but once you see it, you can solve these puzzles almost immediately. 
That's a good example of how thinking mathematically can save work: 
I've done all the work up front in finding a method, and now I can 
solve lots of similar puzzles without having to do a lot of trial and 
error!

Now, I just looked back at your actual problems, and found that there 
is an extra trick to them. Some problems can't be solved, and trial 
and error is completely wasted on them. When you use a method like 
mine, you can quickly see whether the problem can be solved or not. 
And once you realize that some problems of this type have no 
solutions, then you look for a quick test that will tell you which 
can and which can't. There is such a test in this case. And that is 
an even bigger advantage to thinking mathematically: you not only 
save work in solving a problem, but you can avoid doing _any_ work on 
a problem that can't be done!

So look for a reason why some of these might not have solutions.

If you have any further questions, feel free to write back.

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