I don’t usually struggle to distinguish toys from tools. Gas stove? That’s a tool. Easy-Bake oven? That’s a toy. Bricks? Tools. LEGO bricks? Toys.

Mathematical tools are similarly distinctive. They harness industrial-strength power—think of Taylor series, or completing the square. Mathematical tools shine floodlights into dark corners. They unlock doors, solve problems, and make attentive students utter, “Whoa, deep.” They often come with complex instruction manuals, requiring weeks (or months (or years!)) of technical training to master.

Mathematical toys… not so much. They’re simple to grasp, fun to handle, and not much substantive good to anyone. Think of Sudoku puzzles, or differentiating cos(cos(cos(cos(x)))). We might get a kick out of poking and prodding such problems, but solving them won’t teach us anything fundamental about the workings of the universe or the necessities of logic. Toy problems aren’t floodlights; they’re more like flashlights dangling off of a keychain.

But just as the Incas mistook the wheel for a mere toy, sometimes mathematicians get it wrong. Sometimes what seems to be a toy is, in fact, a powerful tool.

Sometimes a toy is just a tool in waiting.

Think of the perfect numbers. Despite the flattering name, they’re not particularly central to number theory. They’re integers whose proper divisors add up to the original number itself. For example, take the factors of 6:

Add them up, and you get 1 + 2 + 3 = 6. Perfect! Or consider 28. Its factors are:

Add them up, and you get 1 + 2 + 4 + 7 + 14 = 28. Again—perfect!

Perfect numbers strike me as a quintessential example of a mathematical toy. They’re easy to explain and recognize, yet surprisingly hard to track down. (The next two are 496 and 8128.) They have no obvious conceptual significance or physical application—only a cute name, a clear definition, and just enough mystery to tickle the mind.

But you can also argue that perfect numbers have the potential to be a powerful tool. They combine two of the most elementary operations in mathematics: factorization (i.e., listing proper divisors) and addition (summing those divisors). That means they occupy prime real estate in the mathematical realm: a special vortex at the intersection of addition and multiplication. So perhaps finding perfect numbers isn’t just a silly puzzle for idle minds.

Perhaps the game really means something.

There are two big unanswered questions about perfect numbers. First, do they go on forever? They clearly grow sparser and sparser (after 8128, the next one is 33,550,336). But a family of numbers can continue forever, despite growing increasingly scarce—think of the primes. So do the perfect numbers constitute an infinite family? Or is there, somewhere out there, a final and largest perfect number, the king of them all?

The second question is: Are there any odd perfect numbers? The answer appears to be no—mathematicians have proven that any odd perfect number would need to satisfy absurdly strict conditions—greater than 10^{1500}, not divisible by 105, and so on. (They’ve stopped just short of proving it would need to float like a butterfly and sting like a bee.) The mathematician James Joseph Sylvester went so far as to declare:

So an odd perfect number would be semi-miraculous. But no one has yet proved it impossible.

The answers to these two questions may shed light on whether the perfect numbers are tool or toy, meaningful or meaningless. Maybe the proofs will reveal deep truths about the integers, a hidden significance to the perfect numbers. Or maybe the perfects themselves will remain a cheap trinket, but the techniques developed to work with them will find applications far and wide—so that the perfect numbers will, in the end, open doors to higher truths.

Or maybe the proof will reveal nothing exciting at all, and the perfect numbers really are just a silly game.

Part of math’s joy—and its frustration—is that it’s hard to tell the tools from the toys in advance. The perfect numbers may look like an Easy-Bake Oven for now. But one day, some clever mathematician may use them to cook a beautiful soufflé.

Thanks to WolframAlpha, I managed to differentiate cos(cos(cos(cos(x)))). It’s sin(x)sin(cos(x))sin(cos(cos(x)))sin(cos(cos(cos(x)))).

WolframAlpha is the best tool of all. 🙂

That’s a toy problem I like to use to illustrate the chain rule’s chain-like workings. It also has the bonus challenge that you have to keep careful track of the minus signs.

Umm, I think you need better analogies. An Easy-Bake oven isn’t a toy, for the simple reason that you can bake real cookies in it. And while people don’t usually make real houses out of Lego bricks, it has been done, though I have to admit that the housecat is strictly a toy cat. Model rocketry is real rocketry in every sense of the term, though no amateur rocket is likely to reach orbit any time soon.

Yeah, it turns out to be surprisingly hard to come up with clear-cut toys that are nicely analogous to real-life tools. I’ll just pretend the ambiguity of my examples was a deliberate prelude to the ambiguity of the mathematical case (even though that totally cuts against the phrasing I used).

So presumably it’s well-known that 2^{p-1}.(2^p -1) is perfect whenever that second factor is a prime; but we don’t know whether there are infinitely many powers of two that are one more than a prime, we just haven’t yet run out of examples. From the lack of certainty (only strong evidence) about odd perfects, I infer that we don’t know whether there’s any perfect outside this (presumably) well-known family. (Your discussion of odd without mention of “even outside this family” leaves me wondering whether someone’s proven that any even must be in this family, or whether we know even less about “even outside this family” than about odd.)

So we actually have (at least) *two* fascinating puzzles about this toy: does the known family have infinitely many members ? and are there any outside this family ? The first is surely part of the nexus of puzzles linked to the Riemann hypothesis (of which one may also ask: toy or tool ?); on the second, despite this family having ever more powers of two, we can’t quite rule out there being an odd example. Which makes both halves of the puzzle deep and fascinating – even though, yes, I totally agree, “perfect” is a silly term for these numbers and they’re nothing but a toy. For now ;^>

Meanwhile, back on the eve of WWII, the eminent mathematician Hardy thought number theory and quantum mechanics were toys (and loved them for it); by the time the war was over, each had made a major contribution to allied victory (yes, your little hammer did tell the truth that first time – the trick to sustained fission was in fact hitting *gently enough*, so a little hammer is the perfect representation of it).