The second part of the Four Fours

The Back Story

The four fours is a rather famous little puzzle which goes something like this:

Using exactly four of the number 4 each time, write calculations using +, -, *, / that produce each of the natural numbers from 0 to 20.

It’s a classic puzzle that requires some creativity and also gets people thinking about how the operations interact with each other. One thing I find both frustrating and fascinating is what happens when people come up with numbers that are very hard to produce with the standard basic operations of addition, subtraction, multiplication and division. The majority of people start branching out into other operations like concatenation (writing 44), decimal points (writing 4.4 and .4), factorials (4! = 24), and square roots (√4=2). Basically anything that doesn’t require the use of another digit tends to be fair game. Indeed, some presentations of the Four Fours puzzle explicitly allow these operations from the outset, and I have been told off for “forgetting” to include them when I present the puzzle myself.

The reason I find this fascinating is that nobody every seems to ask the question that to me is the most natural question to ask about the Four Fours. The first question I asked myself when I was first presented with the Four Fours was “are they all possible with just +, -, *, /?” However, I have yet to meet any students who ask this question or try to figure it out even if they don’t voice it. No, they always ask, “Am I allowed to use [insert operation here]?” I find it fascinating that the first response is to seek out other operations to use rather than to see what can be done with the operations you do have. People seem to be focused on producing the results in any way they can, rather than asking whether it’s possible to produce the results.

I also find it fascinating that once other operations are allowed, it suddenly becomes a game to use the fanciest operations the students know. This aspect in particular is what I find frustrating because for many it’s now a way to show off, and people get praised for a solution using a fancy operation seemingly because it uses that fancy operation. You also start getting solutions using All The Things, even though it’s totally possible to get the answer for some of them just using the most basic of operations. I do get the fun of using All The Things, really, but it does seem to me to go against the spirit of the original problem which is all about how much is possible within constraints.

So here’s the question: how do I arrange the Four Fours puzzle to make it more natural for people to consider what they can or can’t achieve using just the basic operations, and if new operations are allowed, how do I prevent it from becoming All The Things?

The Two-Part Four Fours Problem

And so, I have come up with a two-part version of the Four Fours problem. It goes like this:

The Four Fours, Part 1:
The goal is to write whole numbers from 0 to 20 as calculations, each using exactly four of the number 4, and as many of the operations of +,-,×,÷ (and brackets) as you need. For example, 8=4+4-4+4.
Six of the whole numbers from 0 to 20 are not possible using these rules. Which numbers are they?

The Four Fours, Part 2:
If you were allowed to also use one other symbol or operation or function as many times as you like (along with +,-,×,÷ and brackets), which one could you choose so that you could write all six of the missing numbers as a calculation using exactly four fours?

The first part is presented this way to encourage people to try as much as they can with just the basic operations before trying something else. I deliberately chose to reveal that there were six impossible numbers to remove the need to prove that they were impossible, but If proving was important to you, then you could instead say that some of them are impossible, rather than specifically six of them. Also I wanted it to be very clear when you could move on to Part 2, because in my setting where I work with other people’s students or with strangers walking up in a public place to join in, I can’t always have the luxury of negotiation, which relies on a relationship I don’t have time to form.

It is worth saying that yes presenting it this way does make people curious and keeps them working on it until they have found 15 numbers that are solvable. Well, more people came over and joined in than with some other puzzles we’ve tried at One Hundred Factorial, and more persisted for longer, anyway. Also, when we did it at One Hundred Factorial, people were more systematic than I usually observe with the Four Fours, seeming much more likely to modify existing solutions than to just try random things. Your mileage may vary, of course.

The second part is presented this way to still include the idea of constraint. Sure, you can think of a function/operation that can subvert the rules to get the answers you are missing, but can you make it get all the missing answers? Now you still have to stop to think about the range of possibilities, and evaluate your idea. Plus, I am sharing one of the questions I have had about the Four Fours from very near when I first heard it — I get to see how others respond to a question I’ve always wondered about.

It’s particularly fun to me to look at the list of operations people do tend to allow (concatenation, factorial, roots, decimal points) and see which of them alone allows you to complete the missing six numbers. Some of them actually do allow you to do it, which I find amazing since usually they are all allowed, even though sometimes just one of them is enough.

Some solutions

If you would like to see some discussion of the solutions to Part 2 of the Four Fours, then check out the replies to this tweet. Of course, you may want to try some things yourselves, in which case don’t check out the replies yet!

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