How To Really Solve The Three 3s Problem?

The Three 3s Problem is a fascinating mathematical puzzle set that I stumbled upon recently. As I began solving it, things looked straightforward and I was beginning to get bored. That was until the final leg of the problem caught me out — a little bit of a mathematical plot twist if you will.

At the end of the experience, I thought this problem was well worth sharing. So, here we are. The problem has fairly simple rules. I have modified the rules of the original problem to be more restrictive in this article for two reasons:

1. To reduce the complexity of the mathematics involved with the solutions, and

2. To make the puzzles more accessible to readers.

Without any further ado, let’s jump straight into the rules of the game.

This essay is supported by Generatebg

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The Three 3s Problem — The Rules

As I mentioned previously, the rules are pretty straightforward. All that you have to do is complete the following ten equation sets using only mathematical sign manipulations around the existing three 3s in each incomplete equation.

In order to achieve this, you are allowed to use the following mathematical signs only: ( ), +, –, *, /, and !

Now that you know the rules, I will demonstrate the solution for (arguably) the easiest equation of the whole set. After this, you may choose to solve the remaining equations on your own before continuing to read the essay.

Sample Solution for the three 3s problem

Arguably, the easiest equation to solve is the ninth one which equates to a ‘9’.

The reason is when we add ‘3’ to itself thrice, we end up with ‘9’ as the answer. In other words, all we will need to do is introduce two ‘+’ signs as follows to complete the equation:

Now that you’ve had a sample solution, feel free to tune off of this article and solve the rest of the equations on your own.

Difficulty alert: By far, the hardest problem is the tenth equation.

I spent hours on this one. So, if you feel that you are spending a lot of time on this and aren’t progressing, feel free to skip this particular puzzle and come back to this article to learn the solution.

Spoiler alert: From this point on, I will be discussing explicit solutions to the equations in the article.

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The Rest of the Solutions

I usually approach such problem-sets by treating them in the order of the easiest problems first (seems like such a human thing to do). So, in that spirit, the next equations I’d choose are the ones involving ‘3’,
‘6’, ‘2’, and ‘4’. These can be solved quite simply as follows:

Please note that the solutions that I demonstrate in this article are not unique. They are one among other possible solutions. So, your own solutions might vary.

The equations involving 1, 5, 7, and 8 can be solved using the factorial function as follows:

The Three 3s Problem — Final Hurdle

Now, we arrive at the final hurdle. All that is left is the equation with ‘10’. Like I mentioned before, I spent hours on this one. The solution is not only counterintuitive, but it also involves a concept that not many of you might be aware of. So, I’ll go ahead and reveal the solution first, and then discuss what is going on under the hood.

That is a strange-looking solution, isn’t it? What is the exclamation mark doing in front of a number? Is this even valid? Well, to make sense of this, we need to spend some time understanding the subfactorial function.

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The Subfactorial Function

In combinatorics, the common factorial function gives the number of permutations of any number of discrete quantities. In other words, 3! gives the number of ways in which three quantities can be arranged. If we consider a set of three quantities — a,b, and c, we get the following possible arrangements:

The subfactorial function gives the number of derangements possible for a given ranked set.

Let’s take the example of 3 once again. Given a ranked set of [a, b, c], !3 gives the number of combinations within which none of the set-entities occupy their original position in the ranked set (derangements). We can demonstrate this graphically as follows:

The general formula for the subfactorial function is given as follows:

Making use of the subfactorial function, we do not break any rules of the game, and arrive at an equation that is precisely equal to 10.

Credit and Final Remarks

It is refreshing to come across a novel math function hidden inside a seemingly harmless puzzle. I thank and credit Carl Ho for the awesome puzzle.

As I mentioned before, the original puzzle offered much more mathematical freedom. But I chose to restrict the rules even further so that more people can attempt these puzzles.

Nonetheless, ‘10’ is an odd-ball. It cannot be solved without knowledge of the sub-factorial function (given the current rules). In case you feel a bit cheated, just know that mathematics works that way sometimes. The good news is that we get to learn new and (hopefully) useful concepts after such frustrating experiences.

As is often the case with such puzzles, how one approaches them is more valuable than actually solving the puzzles themselves. In case you came up with interesting approaches to solve these puzzles, please be sure to let us all know in the comments section!

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Further reading that might interest you: How To Solve The Doubling Puzzle? and How To Really Use Mathematical Induction?