If you want to get your own set you can buy one here! Also, if you have any other ideas for games or activities using the cards, please send them my way.
First, you can play a classic game of War. The twist is that while playing you should only look at the diagram side of the cards, not the side with the written-out number. So part of the game is figuring out which factorization diagram represents a bigger number. One could of course just work out what each number is and then compare, but I imagine students may also find tricks they can use to decide which is bigger without fully working out what the numbers are.
Variant 1: primes are wild, that is, primes always beat composite numbers. (If you have two primes or two composite numbers, then the higher one beats the lower one as usual.) This may actually make the game a bit easier, since when a prime is played you don’t actually need to work out the value of any composite number played in opposition to it.
Variant 2: like variant 1, except that primes only beat those composite numbers which don’t have them as a factor. For example, 5 beats 24, but 5 loses to 30: since 30 has 5 as a prime factor it is “immune” to 5.
As a fun follow-on activity to variant 2, try listing the cards in order according to which beats which!^{1}
Alex and his students came up with a fun variant on SET. Start by dealing out twelve factorization cards, diagram-side-up. Like the usual SET game, the aim is to find and claim sets of three cards. The difference is in how sets are defined. A “set” of factorization cards is any set of three cards that either
Here are a few examples of valid sets:
And here are a few invalid sets:
In order to claim a set you have to state the number on each card and explain why they form a set. If you are correct, remove the cards and deal three new cards. If you are incorrect, keep looking!
Alex and his students found that, just as with the classic SET game, it is possible to have a layout of twelve cards containing no set. For example, here’s the layout they found:
Just to double-check, I confirmed with a computer program that the above layout indeed contains no valid sets. As with the usual SET, if you find yourself in a situation where everyone agrees there are no sets, you can just deal out three more cards.
The natural follow-up question is: what’s the largest possible layout with no sets? So far, this is an open question!
Since someone asked in a comment, I thought it was worth mentioning where this comes from. It would typically be covered in a second-semester calculus class, but it’s possible to understand the idea with only a very basic knowledge of derivatives.
First, recall the derivatives and . Continuing, this means that the third derivative of is , and the derivative of that is again. So the derivatives of repeat in a cycle of length 4.
Now, suppose that an infinite series representation for exists (it’s not at all clear, a priori, that it should, but we’ll come back to that). That is, something of the form
What could this possibly look like? We can use what we know about and its derivatives to figure out that there is only one possible infinite series that could work.
First of all, we know that . When we plug into the above infinite series, all the terms with in them cancel out, leaving only : so must be .
Now if we take the first derivative of the supposed infinite series for , we get
We know the derivative of is , and : hence, using similar reasoning as before, we must have . So far, we have
Now, the second derivative of is . If we take the second derivative of this supposed series for , we get
Again, since this should be , if we substitute we ought to get zero, so must be zero.
Taking the derivative a third time yields
and this is supposed to be , so substituting ought to give us : in order for that to happen we need , and hence .
To sum up, so far we have discovered that
Do you see the pattern? When we take the th derivative, the constant term is going to end up being (because it started out as and then went through successive derivative operations before the term disappeared: ). If is even, the th derivative will be , and so the constant term should be zero; hence all the even coefficients will be zero. If is odd, the th derivative will be , and so the constant term should be : hence , so , with the signs alternating back and forth. And this produces exactly what I claimed to be the expansion for :
Using some other techniques from calculus, we can prove that this infinite series does in fact converge to , so even though we started with the potentially bogus assumption that such a series exists, once we have found it we can prove that it is in fact a valid representation of . It turns out that this same process can be performed to turn almost any function into an infinite series, which is called the Taylor series for the function (a MacLaurin series is a special case of a Taylor series). For example, you might like to try figuring out the Taylor series for , or for (using the fact that is its own derivative).
At the time I didn’t know how to prove this, but I did some quick research and today I’m going to explain it! It turns out that determining the value of this infinite sum was a famous open question from the mid-1600s until it was solved by Leonhard Euler in 1734. It is now known as the Basel problem (it’s not clear to me whether it was called that when Euler solved it). Since then, there have been many different proofs using all sorts of techniques, but I think Euler’s original proof is still the easiest to follow (though it turns out to implicitly rely on some not-so-obvious assumptions, so a completely formal proof is still quite tricky). I learned about this proof from some slides by Brendan Sullivan and an accompanying document.
First, recall the MacLaurin series for :
This infinite sum continues forever with successive odd powers of , alternating between positive and negative. (If you’ve never seen this before, you can take my word for it I suppose; if anyone asks in a comment I would be happy to write another post explaining where this comes from.)
If we substitute for we get
Note that the coefficient of is . Remember that—it will return later!
Now, recall that for finite polynomials, the Fundamental Theorem of Algebra tells us that we can always factor them into a product of linear factors, one for each root (technically, this is only true if we allow for complex roots, though we won’t need that fact here). For example, consider the polynomial
It turns out that this has zeros at , , and , as you can verify by plugging in those values for . By the Fundamental Theorem, this means it must be possible to factor this polynomial as
Note how each factor corresponds to one of the roots: when , then is zero, making the whole product zero; when , the becomes zero, and so on. We also had to put in a constant multiple of 2, to make sure the coefficient of is correct.
So, we can always factorize finite polynomials in this way. Can we do something similar for infinite polynomials, like the MacLaurin series for ? Euler guessed so. It turns out the answer is “yes, under certain conditions”, but this is not at all obvious. This is known as the Weierstrass factorization theorem, but I won’t get into the details. You can just take it on faith that it works in this case, so we can “factorize” the MacLaurin series for , getting one linear factor for each root, that is, for each integer value of :
For example, makes the term zero, and in general will make the term zero. Note how we also included a factor of , corresponding to the root at . We also have to include a constant factor of : this means that the coefficient of in the resulting sum (obtained by multiplying the leading by all the copies of ) will be , as it should be.
Now, since we can simplify this as
Let’s think about what the coefficient of will be once this infinite product is completely distributed out and like degrees of are collected. The only way to get an term is by multiplying the initial by a single term of the form , and then a whole bunch of ’s. There is one way to do this for each possible . All told, then, we are going to have
And now we’re almost done: recall that previously, by considering the MacLaurin series, we concluded that the coefficient of in is . But looking at it a different way, we have now concluded that the coefficient is . Setting these equal to each other, and dividing both sides by , we conclude that
Magic!
We also proved that : the product of Dirichlet generating functions is the Dirichlet generating function of the Dirichlet convolution. Now, consider taking in the above definition. We get
which is also often written as just plain . This function is quite famous: it is the Riemann zeta function. The reason it is so famous is because it is the subject of a famous unproved conjecture, the Riemann hypothesis, which in turn is famous because it has been both difficult to prove—many mathematicians have been attacking it for a long time—and deeply related to many other areas of mathmatics. In particular it is deeply related to prime numbers. If you want to understand the Riemann hypothesis better, I highly recommend reading the (truly wonderful) Secrets of Creation Trilogy, especially the first two volumes, which explain it from first principles. (I have previously written about the Secrets of Creation trilogy on this blog: a review of Volume 1 is here, and here is my review of Volume 2). In this post I just want to help you understand a few cool things about the zeta function.
Remember that , so we must have . Also recall that when but it equals otherwise, so in fact
since the only nonzero term is . All together, then, we have
and hence
For example, consider :
This converges to something, although a priori it is not obvious what. By writing a simple computer program, or by asking Wolfram Alpha, we can add up, say, 1000 terms and find that it is approximately . Apparently, the reciprocal of this number is given by
where each numerator is . Again, we can use a computer to check that this sum is approximately : and sure enough, this is approximately !
It turns out that converges to exactly (!!!)—hopefully I can write another blog post explaining that (to be honest at the moment I don’t know how to prove it). We also know that
and there is actually a formula giving for any even positive integer. is called Apéry’s constant, since Roger Apéry proved in 1978 that it is irrational; but we don’t know of any nice formula for it.
I will leave you with a few things to prove that you may find amusing:
Recall that is the number of divisors of , and is the sum of divisors of ; to prove these you will want to reference some facts we proved about and in terms of Dirichlet convolution.
Next time, we’ll see another way to relate the zeta function to prime numbers.
(This might look a bit strange, but bear with me!) For example, suppose for all . Then
(Note that in this case, with , the infinite sum converges; but often we just use as a formal placeholder and don’t particularly care about convergence. That is, is perfectly well defined as an infinite series without worrying about the value of .)
In fact is called the Dirichlet generating function for . Of course, this name should remind you of Dirichlet convolution—and it’s no coincidence; Dirichlet convolution and Dirichlet generating functions are closely related. Let’s see how.
Suppose we have two functions and , and consider multiplying their Dirichlet generating functions (with plain old, regular multiplication):
We have a big (well, infinite) sum of things multiplied by another big sum. By distributivity, we’re going to get a big sum as a result, where each term of the resulting sum is the product of one thing chosen from the left-hand sum and one thing chosen from the right-hand sum. (This is just like FOIL, only way cooler and more infinite-r.) That is, the resulting sum is going to look something like this:
with one term for every possible choice of and . The in the denominator is of course equal to , and we can collect up all the fractions with the same denominator: for each , we will get a denominator of in each case where we pick some and whose product is . So we can reorganize the terms in the above sum, grouping together all the terms where the product of and is the same, and rewrite it like this (is this starting to look familiar…?):
Now we can factor out the (since it doesn’t depend on or ), like so:
But the inner sum is now just the definition of the Dirichlet convolution of and ! So the whole thing becomes
And finally, we note that the thing on the right is itself the Dirichlet generating function for . So in summary, we have shown that
Neato! So in some sense, Dirichlet generating functions “turn Dirichlet convolution into regular multiplication”.
This is far from earth-shattering, but it’s fun to see how a number-theoretic function like arises in a simple formula involving Dirichlet convolution, and how Möbius inversion allows us to quickly derive a related but non-obvious fact. Let’s do a few more!
First, let’s consider . We have
which is just counting the number of divisors of . This function is often denoted . For example, has six divisors (namely, 1, 2, 3, 4, 6, and 12), so . Likewise (1 and 7), (1, 2, and 4), and so on.
The above equation can be restated as , so by Möbius inversion, we immediately conclude that , that is,
For example, we can check this for :
indeed.
As another example, gives us , the sum of the divisors of . This is usually denoted . For example, , , , and so on. Often we also define to be the sum of all the divisors of other than itself, i.e. the proper divisors. Perfect numbers like 6 and 28 are those for which .
Again, since , by Möbius inversion we immediately conclude ; for example, again when , we have
This is a torus, made from 24 crescent-shaped pieces of paper with slots cut into them so they interlock with each other. I followed these instructions on cutoutfoldup.com. There is also a template with some ideas for nice variations here.
The idea of this model is to highlight Villarceau circles. Everyone knows how to find circular cross-sections of a torus, right? You can cut it horizontally (like cutting a bagel in half) in which case you get two concentric circles:
Or you can cut it vertically (like cutting a donut in half, if you wanted to share it with someone else), in which case you get two separate circles:
But it turns out there is yet another way to cut a torus to get circular cross-sections, by cutting it with a diagonal plane:
Note that the plane is tangent to the torus at the two places where the circles intersect. These circles are called Villarceau circles, named after French mathematician Yvon Villarceau, who published a paper about them in 1848. The paper model highlights these circles: each circle is composed of edges from two differently-colored crescents; the points of the crescents come together at the tangent points where two Villarceau circles intersect.
If you want to try making this model yourself, be warned: it is quite difficult! The five-star difficulty rating is no joke. The time from when I first printed out the templates for cutting until the time I finally finished it was several months. Partly that was because I only worked off and on and often went weeks without touching it. But partly it was also because I gave up in frustration a few times. The first time I attempted to assemble all the pieces it ended up completely falling apart. But the second time went much better (using what I had learned from my first failed attempt).
Möbius inversion. Suppose is defined for as the sum of another function over all the divisors of :
Then we can “invert” this relationship to express as a sum over :
The above is the “traditional” way of stating the principle, but I like to write it in this equivalent, more symmetric way:
where, as usual, the sum is over all factorizations of into a product . This is the same as the previous equation, because if and only if , in which case . But now it becomes more clear that we are dealing with a Dirichlet convolution. Can you see how to prove this?
Proof. If we think in terms of Dirichlet convolution, the proof is almost trivial. Note first that saying is the same as saying (recall that taking the Dirichlet convolution with is the same as summing over all divisors). We can then take the Dirichlet convolution of both sides with :
In other words, Möbius inversion is really just saying that if , then , because is the inverse of .
Let’s try an example! Recall that denotes the Euler totient function, which counts how many positive integers less than or equal to are relatively prime to . For example, since all the positive integers less than or equal to (other than itself) are relatively prime to it, and more generally for any prime . As another example, since the only positive integers between and which are relative prime to are , , , and . In a previous post we considered a visual proof that
For example, as seen in the picture above, .
Now, this is in the perfect format to apply Möbius inversion: the sum over all divisors of the function is equal to the function . So we conclude that
or, factoring out ,
which is interesting since it expresses as a fraction of .
Let’s use this to compute . We have
The prime factorization of is . We know that will be for any divisor with any repeated factors, so those all cancel out; we only need to consider divisors which correspond to subsets of . ( is the same example we used in our proof of the key property of the Möbius function.) So
Remembering that yields for an even number of prime factors and for an odd number, we can evaluate this as
(and we can use e.g. a computer to verify that this is correct). Note how this works: we start with and then subtract out all the numbers divisible by (all of them), as well as those divisible by () and (). But now we have subtracted some numbers multiple times, e.g. those which are divisible by both and . So we add back in the numbers which are divisible by ( of them), by (), and by (). Finally, we’ve now overcounted numbers which are divisible by all three, so we subtract off the numbers which are divisible by . Egad! This seems really similar to PIE! Indeed, this is no accident: it turns out that the Principle of Inclusion-Exclusion is really just another kind of Möbius inversion. But that’s a subject for another post!
where the sum is taken over all possible factorizations of into a product of positive integers. Last time we saw that is commutative and associative, and has as an identity, where is the function which produces for an input of and for all other inputs.
So what does the Möbius function have to do with this? Let’s start by considering a different function:
is the function which ignores its input and always returns . Of course this is not the same thing as , and despite its name is indeed not an identity for Dirichlet convolution. That is, if we take some function and find its Dirichlet convolution with , we don’t get again—but we do get something interesting:
The first step is the definition of ; the second step is the definition of ; and the last step just notes that the sum of all where is the same as taking the sum of for all divisors of .
That is, is the function which for a given outputs not just but the sum of over all divisors of .
We’re now ready to see how enters the picture!
Claim: .
That is, is the inverse of with respect to Dirichlet convolution. In my next post we’ll see some cool implications of this; for now, let’s just prove it.
Proof. Here’s the proof. Ready?
Actually, there was hardly anything left to prove! The first equality is because of what we just showed about taking the Dirichlet convolution of with something else. And the second equality is a property of we just finished proving in some previous posts.
The sum is taken over all possible factorizations of into a product of positive integers. For example, suppose and . Then
At this point this may seem somewhat arbitrary, but over the next few posts we’ll see that it’s not arbitrary at all—this operation is deeply connected with the Möbius function and hence with primes and factorization.
So what properties does have?
It’s commutative, that is, . This follows directly from the commutativity of multiplication:
It’s also associative, that is, . This also follows from the associativity of multiplication, and distributivity of multiplication over addition. The proof is kind of fiddly but not really difficult:
Define the function as follows:
Then I claim that is an identity element for , that is, for any function we have . By definition,
But for every other than , which cancels out most of the terms of the sum. The only term that does not become zero is . So the right-hand sum reduces to just , that is, . Since we already know is commutative this proves that too.
So, to sum up^{1}, we have defined the Dirichlet convolution , and shown that it is associative, commutative, and has as an identity element. That is, to use some fancy math words, we have shown that is a commutative monoid over the set of functions defined on the positive integers.^{2} Next time we will see what this has to do with the Möbius function .