But is it math?

It is a persistently popular thing to do on social media to post challenges like this one. I used to be of a mind to be outraged at the abuse of the equal sign: Clearly these are not addition problems! This is not how math symbolism works! This is not math!

However, I’ve since shifted my position (despite instinctive twitches). Yes, these represent an inappropriate use of mathematical symbols, but… does that mean they’re not math? That leads to a bigger question: What IS math?

A standard claim I’ve used myself is that it’s important in mathematics to use communally accepted symbols. But, for one thing, we don’t use universally use the same symbols. Some cultures use commas as the decimal separator, for instance. The linear equation of \(y = mx + b\) is more popular in the United States, while \(y = ax + b\) is used elsewhere. An “apothem” in one area is an “inradius” elsewhere; your “variable” may be my “unknown” and someone else’s “pronumeral”. And what’s the formula for the area of a rectangle: \(A = bh\)? \(A = lw\)?

Surely, though, the addition sign and the equal sign always mean the same thing? Not really. It’s certainly true that, completely devoid of context, \(6 + 4 = 210\) would be properly parsed as saying “six objects combined with four objects is equivalent to 210 objects”, which is factually incorrect. But there’s context here: The implication is that the standard interpretations of the symbols are wrong. Also, the plus sign is used more broadly in some branches of mathematics, and the equal sign can be use in other ways as well (such as for assignment of function results, as it arguably is here).

The question in this meme would certainly be more properly coded by using a different symbol for the plus sign, e.g.:

Given the following statements, determine the effect of \(\heartsuit\):

\(6 \heartsuit 4 = 210 \\ 9 \heartsuit 2 = 711 \)

However, that this is the intended interpretation can be determined within seconds by anyone not being willfully obstinate or obtuse. To comment otherwise is the equivalent of responding to a student’s “Can I use the bathroom?” with “I don’t know, can you?” Yes, it’s abuse to misuse the symbols; yes, it leads to further misunderstanding. I’ve made that argument multiple times in the past. At the same time, that doesn’t mean the underlying problem is wrong, or that it’s not math.

So… is this math?

For a long time, I held that the underlying solution to this problem is not fully mathematical in nature: The simplest version of the solution involves arithmetic, but it also involves string concatenation. The concatenation, I held, would move it out of the realm of plain arithmetic and, by extension, make it “not math”.

One problem with this analysis is that it implies that mathematics must involve the manipulation of numeric values. However, a significant portion of geometry books are dedicated to plain pattern-matching, and predicate logic generally doesn’t involve quantities at all. And while Jordan Ellenberg writes a lot about numbers in “How Not to Be Wrong”, his actual definitions of math are more amorphous than that.

Let’s look at the problem from the context of Common Core’s eight standards for mathematical practice:

1. “Make sense of problems and persevere in solving them.” The realization that this problem, as framed, involves a non-standard use of some symbol (either the plus sign or the equal sign) requires sense-making. The decision that this is not merely an error but an intended part of the problem leads to perseverance in solving.
2. “Reason abstractly and quantitatively.” This, I think, is where there is the greatest tendency to balk. It is possible (as I’ll show below) to create a fully arithmetic function that generates the target values, but what is the meaning of this function? How could it be used in an applied situation? There is a numerical solution, but it’s not a clearly quantitative solution.
3. “Construct viable arguments and critique the reasoning of others.” This is not about the problem per se, but about the discussion surrounding it.
4. “Model with mathematics.” As with #2, it’s not clear how the solution would be used to solve any mathematically interesting problem, either real or abstract. It’s certainly an exercise in inductive reasoning, but that’s about it.
5. “Use appropriate tools strategically.” This is also not about the problem per se; it’s irrelevant.
6. “Attend to precision.” This is the source of my superficial complaints: The problem, as framed, grossly misuses the plus sign. It does not involve trying “to communicate precisely to others”. This example is particularly egregious because it doesn’t even have a question. It simply shows a pattern and instructs people to share if they understand it. It is very imprecise, superficially.
7. “Look for and make use of structure.” This problem is very much about this standard. If the point of the exercise is to find the pattern, and communicate it, then this standard is very much satisfied.
8. “Look for and express regularity in repeated reasoning.” Whether this standard applies depends on how flexible you are. Internal to the problem, the standard applies, but becomes redundant on #7. Problematically, it’s unclear how the solution to this problem can be generalized to solve problems outside of its set; we could derive similar problems of the same type, but what good would that be?

The problem, with its implied goal of finding the pattern, is an inductive reasoning exercise. It can very easily be rewritten into an abstract algebra exercise where the goal is to write a binary function. From that standpoint, it’s a math problem.

It is unclear how the underlying function could be used, or even modified, to solve any problem other than itself. It’s an extreme example of a puzzle that’s only internally relevant, but is that enough to disqualify it from being math? I don’t know. If it’s not math, it at the very least requires “mathematical thinking”.

An arithmetic solution, and more comments

The obvious solution is that what’s to the right of the equal sign is a concatenation of the difference and the sum of the two numbers to the left. 6 – 4 = 2, 6 + 4 = 10. 15 – 3 = 12, 15 + 3 = 18.

To write this entirely using mathematics, we can multiply the difference by a power of 10 in order to “move it” out of the way of the sum. This is a mathematically interesting act: How do we mathematically calculate the number of places held by a number? We can use the floor and log functions: In base \(b\) a number \(x\) has \(\lfloor\log_b{x}\rfloor + 1\) digits, so the arithmetic version of our concatenation is: \[c = (a – b) \times 10^{\lfloor\log_{10}{(a + b)}\rfloor + 1} + a + b\]

What is the domain and range? The implied domain is \(a, b \in \mathbb{N}^* | a \gt b \). Is there anything interesting about the range of \(c\) (the two digit values are {13, 15, 17, 19, 24, 26, 28, 35, 37, 39, 46, 48, 57, 59, 68, 79}*)? If we expand the domain of \(a\) and \(b\) to all reals, or to all integers, or to positive reals, does anything interesting happen to the range?

If we change the function, or the base, does anything interesting happen? What interesting things can we learn about how mathematics works by playing with these parameters?

It’s not easy to see how the problem itself can model a useful scenario, but there are plenty of mathematical games that exist solely for entertainment purposes. This appears to be in that category, and open-minded exploration can hone our expertise with a variety of skills that are extrinsically useful.

* That this series does not seem to appear on OEIS’s extensive database of interesting and not-so-interesting number series suggests that it’s not deemed even remotely interesting, but hey.