“Order of Operations” Puzzles

We have often received questions about things called “Order of Operations problems”, or some similar name. Generally, what that means is simply that they are puzzles to give lots of practice evaluating expressions using the order of operations. I have collected a few quite different puzzles in this broad category.

A free-form expression

We’ll start with a 2002 question that is called an Order of Operations problem, but in fact is just like the 24 game we looked at last time:

Order of Operations Problems - a General Strategy

My homework was really hard. Can you tell me the answer to this problem? 

Make an equation using 7, 26, 46, and 15 to equal 160.

I have worked it out several different times and still cannot figure it out. Here's another one: how can you get 18, 9, 24, and 20 to equal 18? At first I thought they looked easy, but I was wrong. Please help me.

There are two different puzzles here, the first with a large target number, the other smaller.

Doctor Ian answered, starting with an important comment about the best type of help, which is not to just give the answer:

Hi Kara,

For a question like this, the answer is unimportant. The purpose of the question is to get you to generate a lot of _wrong_ answers, since that will give you practice in doing arithmetical operations (addition, subtraction, multiplication, and division). If we told you the answer, it would be sort of like going to soccer practice in your place. What would be the point?

The puzzle is meant for practice in evaluating using the order of operations; to give an answer would remove all its value. But it is also meant to develop perseverance and problem solving strategies, like many puzzles, and training in that is useful. So we will commonly solve a similar problem, sometimes one of those asked about, while leaving the rest for the student.

Doctor Ian first gives a strong hint for the smaller problem, using 18, 9, 24, and 20 to make 18:

In the second part, I can give you this hint: It's possible to use 9, 24, and 20 to make 36; and 36 - 18 = 18.

This is the typical hint we give to someone struggling with the 24 game: Take them past some of the trial and error, and leave them with a smaller version of the same problem. This is the same “last operation first” strategy we used last time; if we make subtraction of 18 the last operations, can we get the required 36 from the other three numbers? This is not hard at this point; give it a try.

Now he explains the strategy itself for the first problem, using 7, 26, 46, and 15 to make 160, which is considerably harder:

That, by the way, is a general sort of strategy you can use.  Pick out one number, and the result, and think about how you could get to the result in one step using that number. For example, to get to 160 from 15, you might do one of these:

  160 = 15 + 145 

      = 15 * 12

      = 175 - 15

      = 2400 / 15

So now you have a set of smaller problems:  Can you use the other three numbers to make 145, 12, 175, or 2400?  If not, you can try again using one of the other numbers as your choice.

The harder problems of this type require a different kind of guess, but these will do for a start. We now have a set of four possible targets for only three numbers, 7, 26, 46, reducing the number of things to try.

… But now that I have explored this problem a little, filling my mind with some of the numbers I can make with pairs from our list, I have a different strategy to suggest: Rather than start with one of the numbers, start with a last operation only. If we suppose the last operation is multiplication, what possible factors could you multiply to get 160? Are any of them numbers in our list? Can any be made from two numbers in the list? How about three? If all these fail, we’d move on to another operation, such as addition. But we don’t need to; this strategy leads quickly to an answer.

Fill in the numbers

Here is a very different one, also from 2002:

Order of Operations Puzzle

What is  _ + _ x _ - _ = 22?

We're given the numbers 2, 3, 4, 8.

Doctor Ian answered with a hint to follow the order of operations:

Hi Laurel,

You're going to do the multiplication first, so the possible multiplications are:

  2*3 = 6
  2*4 = 8
  2*8 = 16
  3*4 = 12
  3*8 = 24
  4*8 = 32

That means that the answer must be one of these:

  __ +  6 - ___ = 22        use 4, 8

  __ +  8 - ___ = 22        use 3, 8

  __ + 16 - ___ = 22        use 3, 4

  __ + 12 - ___ = 22        use 2, 8

  __ + 24 - ___ = 22        use 2, 4

  __ + 32 - ___ = 22        use 2, 3

This is a pretty common problem-solving technique: You replace one hard problem with some easier ones.  

Can you take it from here?

Here it makes sense to consider the first operation first! Here’s an extra hint: We’ll be adding one of the remaining numbers and subtracting the other, so the net result is to add the difference of the two numbers (in some order) to the product in the middle. That can make the calculations quicker.

Did you find the answer yet?

Fill in the parentheses

The next (not explicitly called an order of operations puzzle) is from earlier in 2002:

Where to Put the Parentheses?

I am stuck on where to put parentheses in a math expression to make the expression true. I tried to use the guess-and-check strategy but it wouldn't work for me. Here is a problem I need to put parentheses into to make the equation true:

   9 - 6 + 4 * 6 / 3 = -2

The trouble with mere guess-and-check is that you can’t be sure whether you have missed any guesses. After trying that for a while (which is good to start with, if you’re optimistic), you need to switch over to orderly guesses.

I replied, explaining first why I was going to show the entire process rather than just give a hint:

Hi, Hennaysha.

I can't think of any method you can use other than guess and check; you just have to come up with the right guess. That makes it hard for me to come up with a good hint other than giving you the answer. But I'll go through the problem to illustrate how you might think.

The hope is that there are other problems that will require the same kind of thinking.

You have a negative answer, but there's only one negative sign in the whole expression. If you just subtract 6 from 9, you'll get 3, and there will be no way to make a negative answer, so you must be subtracting some expression greater than the 6 alone from the 9:

   9 - (6 + 4 * 6 / 3 = -2

I only put in the left parenthesis; but wherever I put the right parenthesis, the subtraction will be the last operation done. Or … maybe not! There might be another parenthesis before the 9. It’s important not to make assumptions as we move forward. (In fact, just pondering that possibility just now made me see the solution.)

But where can we put the right parenthesis? We can try all possible places:

   9 - (6 + 4) * 6 / 3 = 9 - 10 * 6 / 3
   9 - (6 + 4 * 6) / 3 = 9 - 34 / 3
   9 - (6 + 4 * 6 / 3) = 9 - 14

Now I’m being systematic; but I may still have more parentheses to insert. These are not the only possibilities.

None of these gives -2; but looking back at them, I see that I can place another pair of parentheses in the first attempt and get the right answer:

   [9 - (6 + 4)] * 6 / 3 = 3  ==>  [9 - 10] * 6 / 3 = -1 * 6 / 3 = -2

One of the things I often tell students is that they need to make a habit of asking themselves, “Am I finished? Is there more I can do?” Here, looking at the expressions on the right, with fewer numbers than the original, made it easier to see what more I could do.

The best "guess-and-check" works like this: you look for ways to restrict your guesses, make them, and then refine those based on the results. In this case, it worked out well to actually simplify each parenthesized version and then repeat the process by looking for places to add another pair; there probably won't always be such an obvious choice for the first pair, but this idea may help at least a little. Mostly it takes patience; if you haven't guessed, this kind of exercise is meant to give you LOTS of practice evaluating expressions with parentheses, so you can expect to have to do a lot of checking.

Now I’ll give you, the reader, a little challenge: What if the number on the right, instead of -2, had been -11. Do you see the new answer?

Parentheses again, with exponents

The next is from 2003:

Where Do the Parentheses Go?

I'm trying to help my daughter with some problems she's been given.  Each one has a string of numbers and operations on one side, and a result on the other side.  What you're supposed to do is insert parentheses around the numbers and operations in order to get the result.  For example:

      2    2    2
 2 + 7  - 3  / 3  - 1 * 5 = 35

I know how PEMDAS works, but I don't see how to solve these problems without just doing a lot of guessing.  Argh!

Exponents don’t really make this much harder, because the notation prevents them from being combined with anything else. The problem as written is “\(2+7^2-3^2\div 3^2-1\cdot 5 = 35\)“. If this had been written in the plain-text form “\(2+7\hat{ }2-3\hat{ }2/3\hat{ }2-1\times 5\)“, it would be possible to do things like “\(2+7\hat{ }(2-3\hat{ }2)/3\hat{ }2-1\times 5\)” (that is, “\(2+7^{(2-3^2)}\div 3^2-1\cdot 5 = 35\)“), which couldn’t be what was intended! So this isn’t quite as bad as it could be. On the other hand, as written we will not necessarily be squaring the 7 and 3’s, because we might have something like “\((2+7)^2-3^2/3^2-1\times 5\)“, changing the base of a power.

I answered, pointing out as we’ve seen that “a lot of guessing” is exactly what it’s all about:

Hi, Lynn.

This sort of problem is just a puzzle -- there is no standard, straightforward way to solve it, you just have to try things out and make intelligent guesses. It may help to read what we have to say about the order of operations

  http://mathforum.org/library/drmath/sets/select/dm_order_op.html 

but there is no specific technique we can give you. I'll just try to get you started, thinking through the problem until I see where to go.

I often do this, just thinking off-the-cuff to demonstrate how I approach an unfamiliar problem without first knowing what the answer is.

Using our e-mail notation, your equation is

  2 + 7^2 - 3^2 / 3^2 - 1 * 5 = 35

Looking at this, I notice that 35 = 5*7, and that there is a 7 and a 5 in the left side. So my first action is to add parentheses so that multiplication by 5 will be the last operation performed; if we can make the rest of the expression equal 7, we'll be in good shape. This is just a guess; it may turn out that we won't want to multiply by 5 at all. But we can try it:

  (2 + 7^2 - 3^2 / 3^2 - 1) * 5 = 35

Here I am starting off with the “last-operation” approach, and first trying multiplication, just as in a previous problem. I tend to do this when the target number is factorable.

Now, it's not obvious that we can get 7 out of that, or that the 7 that is there will in any way show through in the final form (since we can't undo the squaring, and we would have to divide by 7 to leave ourselves with just one 7). But I do see, at least, that 7^2 is much too big, so we have to make it smaller, either by subtracting something big or by dividing.

Size is often a useful thing to focus on, in addition to factors.

As written, the division only affects 3^2; 3^2/3^2 is 1. So we'll want to have parentheses around at least part of 

  2 + 7^2 - 3^2

and we can decide eventually how much of 

  3^2 - 1

to divide by. If we used all of both, we'd have

  (2 + 7^2 - 3^2) / (3^2 - 1) = 42/10

which is in the right ballpark but not a whole number, much less exactly 7. It doesn't help if we divide 42 by 3^2 or by 3 rather than 3^2 - 1.

We needed a big enough number to subtract, and 1 didn’t help. But just subtracting the 9 didn’t either.

Now we can try either adding more parentheses inside (2 + 7^2 - 3^2) to change what we are dividing, or pull the 2 outside of the parentheses.

I might also think backward: If the divisor is as written, 8, then the dividend will have to be 56.

I’ve tried things blindly long enough; sometimes when I step back too long to explain my thinking, I forget to actually think, and miss things. So I sat back and actually tried to solve it.

At this point I've finally looked ahead enough to see the solution. I'll leave you with just a hint: you'll want to take the first choice I mentioned in the last paragraph, and you'll have to change what you divide by as well. See what you can do.

So you should add parentheses into the numerator, and do something else outside. But don’t be fooled and add parentheses into \(3\hat{ }2-1\) to get \(3\hat{ }(2-1)\); that wouldn’t make sense in the original form!

Have you found the answer yet?

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