Problem 616

[stephanbrumme]

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]

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

Good luck

[v6ph1]

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?

[hk]

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

[v6ph1]

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]

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]

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]

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

[traxex]

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.

[beco]

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]

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.

[LilStalker]

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

[hk]

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

[zatalian]

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

[traxex]

Asking for partial results or posting them is explicitly discouraged on this forum.

[JoshEnglish]

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]

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.