Ask Me Help Desk - Number sequences

I am going spare trying to work out this one

A student in a school opens 1000 lockers
The second student opens or closes every second locker
The third student opens or cleses every 3rd locker
Fourth student opens and closrs every 4th locker

And so on, what is the number sequence / pattern

Thanks

OK. Therre are 2 states in which a locker can be: open or closed. On or off. Up or down. ZERO or ONE. This suggests a binary application. I'll give you the first one and half, ask later for more hints - I like this one, it's elegant.

A. 1000 lockers open:

$1000_{base2} = 1111101000$

B. half as many lockers open (500)

$500_{base2}= (up) (to) (you)$

Hi. This question is a classic.

Do you mean that 1000 students do it? This is how the question is normally asked.

I'm not sure where fianchetto is going with his method, but I'm sure we will see in due time

If so, you need to look at what defines whether a locker is open or closed at the end of the exercise.

If, for example, you take any one locker. Let's say locker 432

It starts closed. The only students that will change the state are ones that have a number that are a divisor of the locker number.

Student 1 opens the locker,
Student 2 closes it,
Student 3 opens it,
Student 4 closes it,
Student 5 doesn't touch it (432/5 leaves a remainder)
Students 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, 216, 432 all change the state, these are the divisors of 432.

Now this leaves the locker in a closed state.

We can now see that there is a general rule:

When the locker number has an even number of divisors, it is left closed.
When the locker number has an odd number of divisors, it is left open.

Now you need to find which numbers have an odd number of divisors.

Hint: Most numbers have an even number of divisors, this is due to the fact that when you divide a number by another number, you end up with a third number, the 2 divisors here form a pair of divisors, and most numbers just have sets of pairs.
There is a special set of numbers that have divisors that are'nt paired like this, this is where you will find numbers with odd divisors. Can you think what this set of numbers is?

OK - been working for my boss - he seems to like it better that way and keeps me on.

Sorry about my previous post, I had taken a wrong tack with it. What I wound up doing is determining how many lockers were closed (or open) at each turn (spreadsheet helped), noted the rapidly diminishing oscillation, and proceeded to stability.

I found that the number of closed lockers oscillates a bit then sticks to 448 at turn 26, i.e:

$\lim_{turn\to\infty}(NumberClosedLockers)=448$

Series of closed lockers:

1000,500,499,583,547,549,535,482,523,491,493,490,4 84,471,463,491,487,476,474,465,455,444,441,446,448 ,448,448,448...

Series of open lockers is just the difference from 1000. (approach 552)

I think I like Cap's answer better, though - less labour-intensive :)

Fianchetto, your answer doesn't make sense to me.

Surely every student above 500 will just be closing or opening a single locker, thus it cannot stay at 488 till infinity.

(and in the case where 1000 students have gone, as I have argued and you have agreed, 31 of the lockers will be open, not 552)

I agree with capuchin on this one. This is a classic problem. Since there are only 1000 lockers there will be only 1000 students who open or close lockers. The question is "what will the open/closed pattern be after all students complete their actions". The answer is clearly what capuchin implied. The number of open/closed lockers is not important to this problem.

Inspired by Fianchetto's provably incorrect answer, I set out to find the series myself.

Using only Excel and a moderate amount of intelligence, I have calculated the first 50 terms:

1000, 500, 499, 583, 547, 549, 535, 482, 523, 491, 493, 490, 484, 471, 463, 491, 487, 476, 474, 456, 455, 444, 441, 446, 448, 440, 439, 430, 442, 447, 465, 462, 472, 469, 473, 472, 489, 483, 482, 479, 495, 496, 511, 513, 509, 494, 507, 507, 509, 499

As you can see me and Fianchetto disagree at term 24 (FI:448, ME:440), I'm not sure why, something for you to investigate Fian?

Also we disagree at term 18 (FI:465, ME:456), but I think this could be a Fianchetto transcription error.

The sequence is intriguing, and does not seem to have a more simple pattern than the equation that generates it. (it looks somewhat like a random walk about the mean of 500 at the early stages, as most of the terms are affecting a large number of lockers).

Here are the last 50 terms: (951-1000)

78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31

No suprises here, 31 was the expected last term and the decreasing is expected as each student only affects a single locker, you can see around term 961 (11th term here) the decreasing pattern is broken, this is because 961 is one of the numbers with an odd number of divisors that we talked about. For the OP, there is an extra clue to which set of numbers the numbers with odd divisors belong to here, if you can spot it.

Fianchetto I have the spreadsheet if you want it, it's rather large and unwieldy though.
I hope you find the flaw in your logic! (or the difference in our interperation of the question?)

Quote:

Originally Posted by Capuchin

Inspired by Fianchetto's provably incorrect answer, I set out to find the series myself.

Using only Excel and a moderate amount of intelligence, I have calculated the first 50 terms:

1000, 500, 499, 583, 547, 549, 535, 482, 523, 491, 493, 490, 484, 471, 463, 491, 487, 476, 474, 456, 455, 444, 441, 446, 448, 440, 439, 430, 442, 447, 465, 462, 472, 469, 473, 472, 489, 483, 482, 479, 495, 496, 511, 513, 509, 494, 507, 507, 509, 499

As you can see me and Fianchetto disagree at term 24 (FI:448, ME:440), i'm not sure why, something for you to investigate Fian?

Also we disagree at term 18 (FI:465, ME:456), but i think this could be a Fianchetto transcription error.

The sequence is intriguing, and does not seem to have a more simple pattern than the equation that generates it. (it looks somewhat like a random walk about the mean of 500 at the early stages, as most of the terms are affecting a large number of lockers).

I will provide the last 50 terms below shortly, but need time to extend the spreadsheet (255 columns is not enough!).

I stand corrected - I found the error in my spreadsheet entry, and correcting it have similar results. Including the dyslexic entry:)

Quote:

Originally Posted by Fianchetto

I stand corrected - I found the error in my spreadsheet entry, and correcting it have similar results. Including the dyslexic entry:)

Fantastic! I've put the last 50 terms down too in my above post. :)

Quote:

Originally Posted by Capuchin

*counter-neeners*

But wait... there's more - you just showed faith in my earlier hypothesis that the number of open or closed lockers "oscillates then approaches some stability" - I just got the numbers wrong due to typos in my spreadsheet. There is a saying in medical circles: "When you here the hoofbeats, remember - they may be zebras." You rode your horse (and well, I might add), but also helped me lasso my zebra. "Counter - neeners" accepted graciously, anyway :)

I've generated all the data for this one based on what the two of you were discussing (however far from the actual question it may be :) ) and attached the data and chart. You can feel free to determine the pattern on your own but I wouldn't say that it is anything too significant.

Attachment 2585
Attachment 2586

Just to clear up, I didn't think it would approach stability, but it obviously is completely ordered and predictable after 500. (I know the whole of it is ordered, in that it's generated by a rule, but it last 500 where only 1 divisor is left for most numbers is more ordered)