Help me create a puzzle

I don’t have a good puzzle for you today, and it’s about time for a little math, so I’m making one up. Or working on it. Weird thing about puzzles – you can make up a puzzle that someone has already solved. I think that’s what we’re doing, but let’s try all the same.

Since this is from scratch, we need to develop the topic before the question.

Topic: How many ways can we express a particular number as a product?

How many ways can a number be written as a product? So, what does that mean? First of all, by number we mean “natural number.” Then, what’s a product? For 10, it can be written as – that’s easy, that’s one way. But what about 10 standing alone? For the purposes of this puzzle, I’ll say yes. We will count 10 alone as a “product.” What about ? Nah, I think we should state clearly up front, multiplying by 1 will not count. Which means we really want natural numbers strictly greater than one. Can we have more than 2 factors? Why not? after all, 12 is , that should count. And is different from ? For our purposes, let’s say no, they are the same, they don’t count as two different ways.

Do we have material for a puzzle? Well, let’s list out the first few.

Number	Ways it can be written as a product	Show us
2	1
3	1
4	2
5	1
6	2
7	1
8	3
9	2
10	2
11	1
12	4

So what kind of questions might we ask?

We could ask to have the list extended. Make it up to 20. Or up to 50. Or even 100. And we could ask for observations about what we encounter.

We could ask for the number of ways some larger number could be expressed as a product. Might be fun to try to count for, say, 350.

We could ask for the number of ways some much much larger number could be expressed as a product. If we chose, for example, 792,000, that’s big enough that you might not be able to count by hand, and would have to do some generalizing.

We could ask for some observations. Which numbers can only be written one way? (and why should that make sense?) Is there a relationship between the number of factors and the number of ways? The number of distinct factors?

We could ask for a formula to figure out the number of ways for a number in general. But, as I sit here, I don’t know the answer. And, as I sit here, I expect that the answer might be “complicated” in a way that goes beyond what most kids and much of my audience can handle.

We could ask for formulas to deal with some specific cases…or at least the descriptions of a solution. As a teacher I might have asked for a formula (or description of solution) for powers of 2, or powers of a prime, p. My students frequently confounded me by starting by looking at powers of 10 – but maybe that’s better for grasping some of the complexity here. It might also be helpful to ask for solutions for the product of 2 distinct primes (such as ), 3 distinct primes (such as ) or 4 distinct primes (such as ).

But I’m leaning in a different direction. 2 is the first number with 1 way of being written as a product. 4 is the first number with 2 ways. 8 is the first number with 3 ways. See that pattern? Think it breaks down? 12 is the first number that can be written as a product in 4 ways. What is the first that can be written as a product exactly 5 ways? 6 ways? Do you want to jump ahead? 10 ways? 20 ways?

Some questions

• How many ways can each number from 13 to 25 be written as a product?
• How many ways can 350 be written as a product?
• How many ways can 792,000 be written as a product?
• Which numbers can be written as a product in exactly two ways? Can you show that all numbers with this property can be written as a product in exactly two ways? And that no other numbers can be written as a product in exactly two ways?
• Is there a general formula here? (Let’s leave this out, unless you actually know the answer. I have a guess “yes” but I’m not certain what that guess would be, and I’m aware that it might involve math that I have not studied very deeply.)
• If a number is a product of n distinct primes, how many ways will there be to write it as a product?
• Consider the first number that can be written 1 way, the first that can be written 2 ways, etc, ie 2, 4, 8, 12, 16,…. what are the next five numbers on this list?
• Do you have a question I haven’t thought of?

I’ll try answering these (those I can) during the course of the week.

What do you think?

Is there fun stuff for kids to play with here? For us to play with? Would this make a reasonable activity? (and if so, how?)

And has all the math here been done before? (I’d be shocked if it hadn’t been)

I’m curious what you make of this.

Happy break!