Very Interesting Problem

I just heard of a very interesting problem from my colleague. I would like to think about it some time, but since I have absolutely no time now, I propose it for you, my dear reader.

Let’s assume we have an integer n and an alphabet consisting of five symbols – `1`, `+`, `*`, `(`, and `)`. What is the biggest integer we can write down using no more than n symbols from our alphabet?

For example, if n=3, the biggest integer will be 2 (that is, 1+1); if n=15, my guess – it will be 9 ((1+1+1)*(1+1+1)), etc.

What is seen at once – there is only a sense of looking to just odd integers n because all the legal expressions can be only of an odd length.

So, what can we think about the answer – is it possible to make a function f(n) giving the right integer value depending only on n? Or maybe it is just possible to make a function with some other arguments as well? Maybe it isn’t possible at all to make such a function, but just to make an algorithm constructing such an integer? What do you think about it anyway?

Published in: on Thursday, November 1, 2007 at 5:15 pm  Comments (6)  
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6 CommentsLeave a comment

  1. hi,
    thanks for your comment on my blog.
    Actually I didn’t understand what the problem is!! In fact, I’ve been far away from mathematics world since 5 years ago!!! but happy to see another bilingual blogger 😉
    I wish you best of Luck…

  2. Thanks!
    But I don’t remember my comment in your blog any more.. You have deleted your posts.. why so?

  3. If n=15, I can make 151,807,041

  4. Yes, maybe, but can you make a formula f:N–>N that gives the correct answer for any given n? Or, at least, an algorithm for constructing such a formula?

  5. Sure.

    n=1, 1
    n=2, 11
    n=3, 111
    n=4, 1111
    n=5, 11111
    n=6, 111111
    n=7, 1111111
    n=k, 10^(k-1)+10^(k-2)+…+10^1 + 10^0

  6. Hehe, yes, you are true 😉 I guess, I have to add some constraints again in order not to make it this simple 😉 So – none of `1`s is allowed to follow by another `1`..

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