## Problem 616

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stephanbrumme
Posts: 2
Joined: Tue May 16, 2017 5:00 am
Location: Berlin

### Problem 616

I am still not sure how to understand this sentence:
An integer n>1 is said to be creative if for any integer m>1 Alice can obtain a list that contains m starting from L={n}.
Does it mean:
• n can be converted to at least one integer m>1 or
• n can be converted to every integer m>1
Is it okay if m=n ?

Animus
Posts: 1041
Joined: Sat Aug 16, 2014 12:23 pm

### Re: Problem 616

It means:
n can be "converted" to every integer m>1.

Good luck

v6ph1
Posts: 94
Joined: Mon Aug 25, 2014 6:14 pm

### Re: Problem 616

Do the rules mean for {8} and {9}, the only valid step is to {2,3}.
And from {2,3} to {8} or to {9}
Or are there more steps allowed for the example?
Last edited by v6ph1 on Sat Dec 16, 2017 8:27 pm, edited 2 times in total.

hk
Posts: 9792
Joined: Sun Mar 26, 2006 9:34 am
Location: Haren, Netherlands

### Re: Problem 616

No, there are no more steps allowed starting with {8}, without creating a set you have had already.

v6ph1
Posts: 94
Joined: Mon Aug 25, 2014 6:14 pm

### Re: Problem 616

Thanks.

On my reading, there is no creative number, which can generate all integers >1.
(I have a non-formal prove for this in my mind)
As 0 gets rejected, this is obviously false.

Can anyone post one of the creative numbers?

MHealy
Posts: 28
Joined: Sat Nov 17, 2012 11:32 pm

### Re: Problem 616

I also initially believed there were no creative numbers, so I understand your frustration, but I do not believe it would be right to spoil your discovery of why this is false by providing one (nor, I imagine, do the problem creators, as they did not give such in the problem description).

Good luck!

sjhillier
Posts: 483
Joined: Sun Aug 17, 2014 3:59 pm
Location: Birmingham, UK
Contact:

### Re: Problem 616

v6ph1 wrote:
Sat Dec 16, 2017 8:33 pm
On my reading, there is no creative number, which can generate all integers >1.
If it makes you feel better, I took about two days to convince myself that the answer was not 0. (I did have real work to do at the time too, or I might have been a bit quicker.)

v6ph1
Posts: 94
Joined: Mon Aug 25, 2014 6:14 pm

### Re: Problem 616

Got it - [ removed by admin ]
-> Completely solved: Thanks for the non-hints.

traxex
Posts: 52
Joined: Thu Oct 19, 2017 12:30 pm

### Re: Problem 616

v6ph1 wrote:
Sat Dec 16, 2017 10:57 pm
Got it - and [here's a small hint nobody asked for]!
-> Completely solved: Thanks for the non-hints.
Did you really have to? Nobody other than you has asked for spoilers. People opening this thread expect to see clarifications about what the problem is asking and only that, you gave them an unexpected and unwelcome hint.
Technically, everyone is full of himself.

beco
Posts: 4
Joined: Sat Dec 02, 2017 7:44 pm
Contact:

### Re: Problem 616

stephanbrumme wrote:
Sat Dec 16, 2017 4:55 pm
I am still not sure how to understand this sentence:
An integer n>1 is said to be creative if for any integer m>1 Alice can obtain a list that contains m starting from L={n}.
Does it mean:
• n can be converted to at least one integer m>1 or
• n can be converted to every integer m>1
Is it okay if m=n ?
I am with the same problem.

Is there a maximum implicit?

I mean: n can be converted to every integer 1<m<n ?

traxex
Posts: 52
Joined: Thu Oct 19, 2017 12:30 pm

### Re: Problem 616

beco wrote:
Sun Dec 17, 2017 4:55 am
Is there a maximum implicit?

I mean: n can be converted to every integer 1<m<n ?
No, m can be any positive integer greater than one. So there are infinitely many choices for m.
Technically, everyone is full of himself.

LilStalker
Posts: 30
Joined: Thu Nov 03, 2016 4:32 pm

### Re: Problem 616

Does the development team have the information of how many '0' submission this problem got?

hk
Posts: 9792
Joined: Sun Mar 26, 2006 9:34 am
Location: Haren, Netherlands

### Re: Problem 616

No, we don't keep track of the number and the values of wrong solutions.

zatalian
Posts: 1
Joined: Sat Jan 06, 2018 9:57 am

### Re: Problem 616

Can i ask how many creative numbers there are <= 10.000?

traxex
Posts: 52
Joined: Thu Oct 19, 2017 12:30 pm

### Re: Problem 616

Asking for partial results or posting them is explicitly discouraged on this forum.
Technically, everyone is full of himself.

JoshEnglish
Posts: 1
Joined: Fri Oct 29, 2010 8:54 pm

### Re: Problem 616

I, too, am stuck on the idea that
An integer n>1 is said to be creative if for any integer m>1 Alice can obtain a list that contains m starting from L={n}
implies that a {n} must generate every integer greater than 1.

By my reading, 4 is creative because {4} can become {2, 2} and satisfy the condition that this list contains "any integer m>1".

traxex
Posts: 52
Joined: Thu Oct 19, 2017 12:30 pm

### Re: Problem 616

According to this Wikipedia article listing mathematical symbols, $\forall$ can be read as "for all", "for any", "for each", or "for every".

To me it was very clear that the meaning of "for any" in this problem is $\forall$, "for all", "for every"; rather than $\exists$, "for some", "there exists".

On the other hand, this answer on StackExchange gives some examples where the meaning of "any" is less clear, and one comment mentions a book on mathematical writing that discourages the use of "any" because of possible ambiguity.
Technically, everyone is full of himself.