To find a solution to a graph, you must first number your friends (the big circle nodes of the graph) with consecutive integers starting with 1. Here is an example:

One example…

Another example…

A third example… Now we start adding non-negative integers to the edges…

Let’s start by adding a 1 to the edge on the right. That makes one of our friends happy…

How are we going to make our #3 friend happy?

By making the other edge going into it a 2. 2+1 = 3 so now our #3 friend is happy.

Let’s now add a 2 on the edge connecting the 5 and 2…

How are we going to make our friend #2 happy. They have a 2 edge going into them already so the other edge must be a zero.

Good. Now the #2 is happy. What number should we put on our remaining edge?

That was a trick question (you should do lots of trick questions in your classroom!) The graph fails becuse of a fussy friend #4 if you put a 3 on that remaining line…

…but the graph also fails if you put a 2 on the line because now your #5 friend is fussy.

Can this graph be solved without having a fussy friend? Follow these steps:

1) Choose which big circles to put your friends 1, 2, 3, 4, 5, 6, and 7.

2) Put non-negative integers on the edges.

3) Check to see if each friend is the sum of the edges connecting them to the group.

I tried this way to position my friends, but it’s impossible to solve without having a fussy friend. (You can prove it quite easily by realizing that the only way to make 7 happy is to fill its edges with a 1 and a 6…)

The #7 friend is happy and so is the #6 as long as we set its other edge to a zero.

But now we cannot make #5 happy!

Lucky for our friends, there are other ways to organize them around the circle.

I tried this way. If you are an elementary school teacher try solving this and other loops. Not all loops are possible. Indeed, all 5 and 6-friend graphs are so difficult that they should be reserved for especially quirky students. (If I’m honest now, I’d tell you that they are all impossible!)

…but this 7-loop is possible to solve!

This one is possible too…

Not like this… but it is possible. It likely has an awfully large number of solutions. I’ll show you just one on the next page.

Which graphs have solutions? Can you make any generalizations? Please write me, and I’ll include your ideas here. Elementary school teachers should stop here.

If you are teaching algebra, this is a great puzzle to give your students some fun practice. Let’s assume the lower edge gets an x. What other edge do you know something about?

You actually know something about the edges on either side.

You can continue around the whole circle…

Simplifying as you go…

So what value must x be? Hint: Look at the #1 friend at the top.

Of course, x must be equal to 4.

Tell me all about your discoveries! I cannot wait to hear them…