Expressions Challenge

The Four Fours

In my first year of teaching I came across the Four Fours challenge. I loved it. I’ve always considered puzzles, brainteasers, and trivial challenges of that sort to be rather entertaining, so I went to town on the Four Fours right away. The problem gets better as you go: easier target numbers are out of the way and the real mental effort begins as you look to fill in the gaps.

Beyond the personal puzzle perspective, as a brand new teacher (my course load then was split 50/50 between 7-8 and 9-12) I considered this challenge to be a gem for my middle school students. Here’s a few reasons why:

  • Mundane-yet-necessary practice is made more engaging in the context of a challenge (just like in this problem).
  • Shifting the task from evaluating expressions to writing expressions ramps up the critical thinking component of the task. Students have to look for and make use of structure in order to bend the expressions to their will.
  • The problem is flexible and can be presented to individuals, small groups, entire classes (class vs. class challenge, anyone?), or with a twist (find as many expressions as possible with a value of, say, 23).

And Then It Happened: The Internet Failed Me

I’m sure there are more reasons why the Four Fours is awesome. If you think of any, share them in the comments. But I know of one reason why the Four Fours is not awesome. In fact, I know of one reason why the Four Fours is worthless. That’s right, useless. Totally devoid of any value.

Wait? Just a minute ago I was singing the problem’s praises. Why the dramatic turn? One reason:

The Internet

I typically love the Internet. Defender of the integrity and usefulness of Wikipedia and all that. But not this time. Here, the Internet failed me.

See, someone decided it woud be a good idea to ruin the Four Fours by posting entire solutions all over the place. Now any time I give this problem to students, they’re one Google search away from Four Fours glory.

My Solution

So what do you do if the Internet robs you and your students of a great problem? Make another one. Keep the good stuff (see the bullets above) and tweak the parameters.

Flash back to my first year. When someone told me the full solution to the Four Fours was online, I rewrote the rules and called it Expressions Challenge. Once I got going, it was easy to make additional versions. I’ve included the directions (and handouts I mocked up this year) for the first three versions, as well as some comments on how to tweak Internet-proof the problem further.


Here are the directions for the first three versions of the Expressions Challenge. Note that there is only one difference between the three: Use the numbers in (1) any order, (2) increasing order, or (3) decreasing order. Obviously, versions 2 and 3 are more challenging.

expressions.1 expressions.2 expressions.3


Just to be clear on the order element of the directions, here are examples for each version (1, 2, and 3, respectively):



And the handouts (nothing special):

Further Defense Against the Evils of the Internet

If the Internet destroys any of the challenges above, no problem. Just tweak the problem again. Use the numbers 1, 2, 3, and 4. Or 1 through 6. Or the first four odds (or evens). You get the idea.

Also, you could change the word “numbers”  in the directions to “digits” to open up some additional possibilities. And for the record, whether your students should be allowed to use decimals, percent signs, radicals, ceiling/floor/rounding functions, etc., is totally up to you.

Let me know in the comments if you find any of this useful, or if you have a similar challenge you use with your students.

Comments 7

  1. I did a similar “clock” exercise with my 7th graders, Three 3’s. They had to form the numbers 1-12 using only three 3’s. I asked then to not go in the Internet… I think most complied! 10 wastage hardest, then 8. They had fun (I think/hope)!

  2. I like that this sort of challenge can be a large scale group challenge, or a bite-size task for individuals or small groups. Thanks for sharing the “clock” version. I think I’ll use that with my Pre Algebra students in the next couple of weeks.

  3. When I was student teaching I first learned about a challenge similar to this using the current year (it was 1998 ouch!) it got tough though in 2000, so I switched to 1234 (numbers must stay in order) I usually do this in the fall as a competition between classes, the idea being to be the first group to figure out all the natural numbers up to 100.

    The last couple times I did it I would add new operators after a couple of weeks to keep the energy up and since the kids ran out of solutions.
    I used an arrow –> to mean concatenate, I always allowed things like 12 + 3 + 4 but now you could do (12 – 3)–>4 to get 94.
    One year I also let in the greatest integer function.

    Now that it is 2013 I bet you could get a large number of answers using the year again. I will try this in the fall.

  4. Thank you so much for creating the Expressions Challenges! I love the Four Fours, also, and was sad to see the solutions so readily available.

  5. Another idea is to keep four 4’s, but to have them work on negatives. I haven’t seen those answers on the internet.

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