Solving the plus sign problem

How many addition signs should be put between digits of the number 987654321 and where should we put them to get a total of 99?

This is a fifth grade problem taken from Word Problems for Kids by Canada's SchoolNet.

The first step, as always, is to understand the problem. The student needs to know what is an "addition sign" and a "digit". We're simply asked to put plus signs in between those numbers and add them up, and try to come up with 99.

Then, after we have a basic idea of what the problem is about, is the time to do something. You know, often the child may say, "I don't know how to start. I have no idea what to do!"

But in this case, as often happens, you'll get somewhere as soon as you'll do something. It's really simple: put some plus signs in there and just see what happens. Let's simply put the plus sign in between every digit:

9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45.

Great, we got something. We got 45 which is too small. Now the student should start thinking HOW to make the sum bigger?

Obviously, the only way to do that is to use ONE OR SOME two-digit numbers. We need to omit at least one of those plus signs!

So try something. For example:

98 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 136.
9 + 8 + 7 + 6 + 5 + 4 + 3 + 21 = 63.

Encourage the student to try a few other possibilities. He/she should notice that if you make the two-digit number using the large digits, the sum will be MORE than if you use the smaller digits for that.

(That is, of course, really simple.)

At this point the student might simply use "brute force" and write out all the possibilities with one two-digit number. That's alright; that will get him an answer!

However, there is also a quicker way, if we use some thinking. The target answer 99 is about half-way in between the two sums I have above (136 and 63), so I would try next to use the "middle digits" such as 4, 5, and 6 as the two-digit number:

9 + 8 + 7 + 6 + 54 + 3 + 2 + 1 = 90, which is too little.
9 + 8 + 7 + 65 + 4 + 3 + 2 + 1 = 99, which is the right answer!

Any other possibility with one two-digit number will either be less than 90 or more than 99, so that is the only solution using one two-digit number and seven plus signs.

But, good problem solvers will also consider checking if this is the ONLY solution. There is the possibility of using TWO two-digit numbers. Indeed, I quickly stumbled upon another solution that way:

9 + 8 + 7 + 6 + 5 + 43 + 21 = 99.

It's easy to note that this is the only possibility using six plus signs, since if you put the plus sign between some other digits, your sum will be MORE than this sum.

I hope this is helpful to some of you. I know the problem is quite easy. My intention is simply to point out how a typical problem solving process can go. Observing the "tricks of the trade" can help you to solve problems, and to teach others do the same.