Impossible? Try anyway!

(An archive question of the week)

Here’s a little problem with some big lessons for problem solving.

Not enough equations?

This question is from Kenanga in 2017, not long before the Ask Dr. Math service stopped taking questions:

Insufficient Information In Question

Given

   A + B + C = 1
   B + C + D = 2
   C + D + E = 3
   D + E + F = 4
   E + F + G = 5
   F + G + H = 6
   G + H + I = 7

Calculate: A + E + I = ?

There are 8 variables but only 7 equations.

One of the things you learn in solving systems of equations is that when there are fewer equations than variables, the system will not have a unique solution. But this is not your typical problem!

Doctor Ian answered, starting with what makes it possible to solve:

Hi Kenanga,

It's true that you have more unknowns than equations, but you're not being asked to find the values of all the unknowns. You're being asked to find the sum of three of them.

Ir may well be that, although there are infinitely many solutions to the system, the value of \(A + E + I\) is the same for all of them. (In fact, since the problem was given by a presumably reasonable person, there’s a good chance of that. If it was made up randomly, or arose from a real problem, you might not assume it can be solved.)

Making the problem more visible

Next, a useful way to make the nature of a system of equations more apparent:

Have you tried aligning the unknowns in those equations?

   A + B + C                         = 1
       B + C + D                     = 2
           C + D + E                 = 3
               D + E + F             = 4
                   E + F + G         = 5
                       F + G + H     = 6
                           G + H + I = 7

That at least makes it easier to see that A and I appear in only one equation each. That might turn out to be helpful. Or it might not.

So A and I are special; is E, too? All we can do is try something:

If we add all the equations, we get

   A + 2B + 3C + 3D + 3E + 3F + 3G + 2H + I = 28

If we add the 1st, 4th, and 7th equations, we get

          A + B + C + D + E + F + G + H + I = 12

        A + E + I + (B + C + D + F + G + H) = 12

So if we could find the sum that I re-grouped in ()'s, we'd be golden, right? I don't know if we CAN do that. It's not like I already know how to solve this! I'm just playing around here, to give you a sense of what kinds of things you can look at.

Why choose those three equations to add? Because they involve, respectively, \(A + B + C\), \(D + E + F\), \(G + H + I\), which together include all the variables once each. We’re using one of the patterns revealed by the aligned version of the equations.

What if …

Now what? I can see several things I might have tried, such as subtracting this equation (total 12) from the other equation (total 28). But he has a goal in mind, and sees a way to get to it:

What if we add the equations that DON'T feature A, E, or I? 

              B + C + D                     = 2
                              F + G + H     = 6

That gives us 

              B + C + D   +   F + G + H     = 8

That's just what we were hoping to find!

It’s just a small step from here; like Doctor Ian, I’ll leave that for you …

Some lessons for problem solving

So now we have an answer, and it would be tempting to just move on, but that would be a mistake. There are some important lessons to be learned here.

Don’t let rules of thumb stop you

The first is that just because you expected to start with more information, that doesn't mean that a solution will ultimately require more. In particular, rules of thumb like "You want to have an equation for each unknown" are just that -- rules of thumb. They're mostly true, but sometimes they're not.

Expectations are not absolutes. When your expectation is about something slightly different from what you are faced with, expect that the result might be slightly different, too.

Focus on the actual goal, not your expectation

The second is that it's important to keep in mind what you're being asked for, and not inadvertently think that you have to find something else. In this case, it's intuitive to think that in order to find the sum of A, E, and I, you first have to find their individual values and then add them up. But that's not the case, is it? You were asked to find the sum of three unknowns, and by looking for that directly, the lack of an eighth equation didn't stop us. For that matter, if we did start with eight variables in eight equations, we may have been tempted to plunge into a lot of tedious (and error-prone) algebra.

You’re used to problems where you have to find each variable, so you expect to do that in order to answer the question. But interesting problems may involve shortcuts.

When you can’t think, play

The third is that, when you don't know what to do, the worst thing to do is just throw up your hands and do nothing. Try things. Play around! See what you can conclude from what you've been given. Try to establish what you would need to know to be a step or two from a solution -- like working a maze backward as well as forward. That's what happened here: we saw that if we could get the sum of the OTHER unknowns, we could get the sum we were looking for. And that suggested a strategy: look at the equations that have ONLY those other variables.

Playing with a problem is a fundamental mindset for solving problems. If you don’t see how to open a puzzle box, turn it around in your hands to see it from different perspectives. Imagine what it might look like opened, and let your  mind wander into different ways that might happen. Then try pushing in various places, to see what happens …

Working backward and forward at the same time is one of the strategies we saw in a recent puzzle example (“bidirectional search”), and it worked well here, too.

Never give up.

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