How Many Triangles Have Sides Whose Lengths Total 15?

Date: 6 May 1995 17:38:58 -0400
From: Dana Murray
Subject: Question

How many triangles can one construct with integral sides adding up 
to 15 ?

Thank you.

Date: 6 May 1995 21:52:48 -0400
From: Dr. Ken
Subject: Re: Question

Hello Dana!

What a neat problem!  I think the first thing to do in a problem like this
is to take a survey on what our tools are, and how we should attack the
problem.  The way I see it, our tools are the Triangle Inequality and the
partitions of 15.  (just to make sure you're still with me, the Triangle
Inequality says that the sum of the lengths of any two sides must be 
greater than the length of the third side; this rules out possibilities like
1,1,13.  The partitions of 15 are simply all the combinations of positive
integers that add up to 15, like 1,1,13 or 2,13.  Obviously, we'll only be
using the partitions that use 3 positive integers)

So let's find a method.  First think about how many triangles we can make
using a side of length 1.  Well, since the Triangle Inequality says that 1
plus the second side has to be more than the third side and vice versa, the
only triangle that will work with a side of length 1 is the triangle 1,7,7.
The ONLY one.  So from now on, when we're looking for triangles, 
we don't have to worry about sides of length 1.

Now how many triangles can we make that have a side of length 2?  
Again, use the Triangle Inequality to tell you that 2 plus the second side 
is greater than the third side, and vice versa.  The only numbers that will 
fulfill that requirement are 2,6,7 and 2,7,6, and since you sent in the 
problem, you get to decide whether or not those are the same triangle or 
two different congruent triangles.  

Then you can keep going like this, and each time you deal with a side
length, you get to throw it out and not consider it for any of the rest of
the problem, since you've gotten ALL triangles that have it for a side
length.  One more hint: what is the _longest_ a side can be in one of these
triangles?

-K