Thread by @edsouthall

So this set of questions is the result of a few days thinking about prime factors, and 15 pages or so of notebook scribbles. I've had more DM's about these questions than anything else I've written so... ANOTHER THREAD.

https://twitter.com/edsouthall/status/1356726290947178497

Firstly, it's criminal that people 'get through' prime factors in like, 2 lessons max. This topic should be *everything*. It's amazing and gives a billion insights into multiplication and division.

So the prime factor 'tree' is nothing new, but if you unroll it like the image on the right, you miss all the subtleties of what prime factorisation tells you, and can help you deduce. The image on the left is FAR more useful and clear.

Now, if you think about the concept of multiples and factors, it stands to reason that a multiple of a number has ALL the factors of that number as its own, with some new ones (the 'multiple' bit) eg if it's a multiple because you x 3, then there's an extra prime factor of '3'

So if we work backwards from that idea, then it becomes clear(ish) that you can determine all the factors (not just prime factors) of a single number just by studying its prime factors - because your original number is just a multiple of all of its factors. Hence...

You can solve Q2:

Now, furthering that idea, if you study the prime factors of a square number, you might notice something. Take a simple number like 9, its prime factors are 3 x 3 or 3². We know 9 is a square number, and its root is 3. 9 is 3², and its root is 3...🤔

now consider a bigger square number, like 324. Its root is 18, and its prime factors are 2² x 3⁴. Well we can rewrite its prime factorisation as (2 x 3²) x (2 x 3²). In other words,

(2 x 3²)²

and 2 x 3² = 18.

Spotted it yet?

We can identify an integer square root from prime factors! We just have to check whether the prime factorisation can be written as "a x a", then the root will just be 'a'. Same for cube root (rewrite prime factors as "b x b x b", cube root is "b" etc. So now we can do these!

eg for Q3, if the pfs are 2⁴x5²x7, then a factor is (2x2) which is 2², so that's one square factor, another is (2 x 5) x (2 x 5) or (2x5)² so that's another one etc.

for Q5b, 900 = 2² x 3² x 5² = (2 x 3 x 5)² so it has an integer square root which is (2 x 3 x 5) = 30

Q1, 6, 7, 8 took a lot longer to write and finalise. I wrote the others by thinking about factor trees, but it was only when i had a moment of inspiration to think about Venn Diagrams that the rest came together. It's my son's birthday today so more in a little while on these.

ok, boy has presents, boy full of cooked breakfast, let's continue...

I think it's important to note that a few dead ends were pursued, I don't just have some innate ability to spot a good question without any prep or pre-planning. For example, I thought about doing some kind of consecutive numbers thing, but it didn't really work.

Note that I start with an example of what the answer could be, and work backwards from that. That works well sometimes, but also other times it can totally sabotage your thinking as you'll see in a minute.

Another avenue abandoned was some kind of 'two square numbers sum to equal a triangle number with two prime factors' question. I think there's only two possibilities <100, and one of them involves the number '1', but that's kind of problematic as 1 is square AND triangle,so is 36

Anyway, the point here, is that i want people to know that we all come up with a load of ideas, follow them to their end, or until you see that their end isn't great, and move in a different direction. That's totally invisible in finished products / solutions but its important

I guess the other thing to note, is that a lot of 'harder' questions involve thinking about extra conditions, involving (in that example) triangle numbers, or 'odd / even' clauses, or '<100' restraints. They help a general solution become a specific one.

I'm getting to the Venn stuff, I promise... just... slowly!
The other two ideas I scrapped were LCM of things like a²x + 3bx³. Nowt wrong with that, it's just very kind of "A Level question", so it felt meh.

And "What is 24310 ÷ 187". I think that's a great q but like this tw-

-eet I ran out of space.

Here's a solution to that last one so you can follow my random brain a bit better!

Anyway, VENN DIAGRAMS. They're both the worst and best. Worst because I've seen and been guilty of just ploughing through to these and basically saying 'so do the venn and bingo!" (no-one gets why it works or what it's doing), and best because if you study it, you get rewards

So look again, and ask yourself, how much information in here can I remove and still deduce? Can I identify all common factors (not just prime factors) from this? Can I deduce uncommon factors (not just prime)? How would this look different if I used a multiple of 252 instead?

oh, and 'will my number, or any factors, be even? If so, how many?' that's a fun one, as it all relies on any '2's.

Anyway, i looked at this kind of Venn setup and thought 'use the Venn, Luke', which is weird, because my name is Ed.

So then I just played around and started building up ideas. Like here: Assume there's a bunch of prime factors missing, but what do we know about them already (the left one is going to be even, that's undisputed). Ok, so what if I said the right is even too? What has to happen?

well, not only does the right circle now have to have a '2' in it, but because the '2' is on the outer region on the left, then the new '2' has to go in the *middle*. Mmmmm puzzley

constraints, cause and effect thinking, it's all coming together. So the 'shared prime factor' element is super interesting from a question writing perspective, because it opens up the 'cause and effect' thinking thing. So let's look at question 8

We know that all the numbers present in the Venn (in the solution) must multiply to equal 540, but we don't know where they go...

540 = 2² x 3³ x 5.

So any combination of those in the Venn right?

Wrong! ...

The fun part of this q is that the first '2' is already in A, so that 'cause & effect' thing kicks in when you're trying to place the 2nd '2'. It can't go in the outer region of B. Hence the q is more interesting than something where say, a factor is missing & it can go anywhere.

omg this thread is enormous sorry. Onto this q. The algebra thing is a pretty standard tactic to make a number based q a bit more challenging, but this brings back my earlier point of the danger of working from the answer... I started with something like the image on the right:

but by having an answer already, it clouded my thinking into assuming there was *only one answer*, when in fact, without a further constraint (y<20) there are infinite (oops). I should point out my sparring partner here was clever clogs @JusSumChick who guinea pigged these first.

Now, with all those brain workings and tactics in mind, I recommend you attempt these yourself and see if you can unpick why they are designed as they are. eg, why shade the middle region in the Venn? Why have the 'even' constraint alongside the '<100' in Q1?

Final point: Good questions take a lot of planning. It's very easy to look at something that exists and think it's simple, but it's the making it exist bit that's hard! Also, the fact that puzzle writing is itself a puzzle, is what makes it fun for puzzle writers.
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