Exciting maths question - really!!

[GooseyLoosey]

For the first time ever, I found myself really excited by ds's homework and even more excited by the answer. I actually did end up understanding how maths can be beautiful.

This was the question: you have a corridor with 1000 doors and 1000 people lining up to walk down the corridor. Person 1 opens all of the doors, person 2 shuts all the doors that are a multiple of 2, person 3 touches all of the doors that are a multiple of 3 and shuts the open ones and opens the closed ones. Person 4 does the same for multiples of 4 and person 5 for multiples of 5 etc. all the way up to person 1000 who shuts the last door.

How many doors are left open after the 1000th person has walked down the corridor?

I loved this question, I really did so I thought I would share it with anyone else out there who likes maths.

Yes, lovely question. I'll try it on DCs tonight. What year is your DS?

Yr 5 - tbh he struggled with it but I was captivated by it!

Is it only the prime number doors that stay open? They'd be the only ones that wouldn't get shut by a multiple of 2, 3, 4 etc.

No, it's the squared numbers that will be open at the end, as they have an odd number of factorials (ie. for 4th door, opened by 1, closed by 2 and opened by 4, then left).

So how many squared numbers are less than 1000? 31... as 31 squared is 961 and 32 squared is 1024.

Am I right????

That's it Jenny - it's the simplicity of the solution that got me. It's elegant somehow.

I love this problem too. I teach it that it is a gaol with gaolers and criminals, so which criminals get to go free at the end.

Oh good GOD, will my Y5 be bringing stuff like that home soon? I'd be frigging terrified of trying to work that out!

Someone please explain it simply? I didn't realise squared numbers came into the curriculum this early?

Unrequited - factors usually come in pairs so most numbers are divisible by an even number of factors (which would mean that the door with that number is shut). By way of example, the factors of 12 are: 1 x 12 which makes a pair with 12 x 1, 3x4 with 4x3 and 6x2 with 2x6. This is even true for prime numbers - so 1x17 and 17x1.

The only numbers that have an odd number of factors (and so the door would be left open) are square numbers, take 16 for example: 1x16 with 16x1,2x8 with 8x2 but there is no pair with 4x4 and so 16 has 5 factors.

Not sure I explained that with the elegance it deserves but hope you get it.

Wouldn't worry about the homework - this was given to ds in a "see if you can do it" kind of way (also, Google is your friend - just type in 1000 door problem).

My brain has just leapt out my ear and legged it...

my 11 year old maths geek did this last night after I found it on here

she's taken it to school to challenge the teacher (didn't have the heart to tell her he is probably aware of it already)

Would love to know what the teacher thinks sausage.

Like the idea of doing it with prison cells.

I think I am wierd, but even thinking about this makes me smile!

thank you for the explanation! Very interesting and very helpful!

I'm going to sit down and crack it. Tomorrow.

Ooh adding this to my lesson tomorrow. Thanks

Ok, I get the question and understand the answer, but I cannot imagine my ds being able to work this out. (I could not have worked this out myself)
He is good at maths and in the top stream etc.

Maybe I am underestimating him. I will give it to him tomorrow!

Steppemum - DS did it by doing a table with doors 1-20 and then working out what the pattern was. .he couldn't do it in the abstract.

Will be using this to stretch an able child in my class. Thanks!

forgot to ask him today, must ask him tomorrow, very interested in if he could work it out. I am amazed at the things he 'sees' that i don't

If he likes it, I will get him to take it in to his teacher - see if she can do it!!

He did it!
we were eating dinner and I gave it to him. After some discussion, dh said 'so would door 1 be open or closed?
What about door 2? and door 3? and so on.

I repeated the pattern back as we they worked their way down the list.

When we got to 12 I recapped ' so door 1, 4, and 9 are open. ds thought about it for a minute and then said ' next door open is 16!!!

dd1 (aged 8) was doing it with us, she got it, but was slow to work it out (realised her times tables are a bit shaky)
She couldn't work out the total number of doors open, but ds did. He narrowed it down to between 30 and 35 in his head but had to get a piece of paper for the 31x31 and 32x32

Very nice problem, we loved it

(got a bit sidetracked at first about our prisoners escaping through the doors once number 1 had open them, so the closed/open doors then become irrelevant.....)

These always sound so much worse when you read them than when you draw them or see them on paper! Word-logic problems are so evil!
Squared numbers are also called indices. They're actually quite a fun way to take a new look at the times tables. Going over the 2x, 3x, 4x, 5x, 6x etc can get so boring!
The list of squared numbers is a fun one to learn as it slots into all the tables
2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8 and what have you. Anything to stave off boredom!

Ooh - thanks A Teacher.
I've gone over all the time tables with my class and while some of them know all of them, I have lots who don't. But this will give me something to with all of them as a starter this week.

Nooo A Teacher. Squared numbers ate not called indices. The index (plural indices ) refers to the little 2 sitting above and to the right of the number you're squaring. You get indices when your cubing or taking 4th powers etc too. Sqauared numbers are called squared numbers or squares or second powers if you're feeling long winded. I'm guessing that the thing you teach isn't maths beyond primary age.

I apologize profusely for the grammar crimes, particularly the incorrect "your". I blame autocorrect. :)

Haha! No. I teach English. & please do correct. I'd rather know what's right and what isn't :-)
I lump powers of two with indices because I'm doing indices at A-Level and the power of two is called an index there: I guess at secondary they call it powers?
Please feel free to correct, and no I won't pull you up for grammar! I'm trying to teach my own kids maths and it's working so far, mostly, I think because they're loving the attention rather than my incredible Maths skills. LOL I have to say, getting up at 6am to do the A-Level work is really testing my sanity!