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Which is the biggest 8 digit number of the form "abcdefgh" which is made up only of 1, 2, 3 and 4 and which follows the rule: the digit 1 is one digit away from another 1, the digit 2 is two digits away from another 2, the digit 3 is three digits away from another 3, etc.?

Noifma
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    So the $1$'s are adjacent, or they have a number between them? Edit if you can't comment yet. –  Apr 29 '14 at 17:57
  • And do you mean "exactly one digit away" or "at least one digit away"? And can we use each number how ever many times we like? You need to be more specific. –  Apr 29 '14 at 18:03

2 Answers2

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Assuming that the conditions are like $1 - 1,2--2, 3---3, 4----4$, I suggest $$41312432$$ Starting from putting the 4's you get something like that $$4----4--$$ then you should maximize the number, so work on the 3's. The first 3 can only be put in the third place to get $$4-3--43-$$ Now the conclusion should be obvious, since a 2 must be put in the last place and the 1 in the second to get $$41312432$$

sirfoga
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  • $24212$ has $2-2-2$ twice. –  Apr 29 '14 at 18:04
  • hmm, that could be a problem... :) I'll edit – sirfoga Apr 29 '14 at 18:05
  • What about $43121432$ if we just need "at least" that many spaces not exactly that many? –  Apr 29 '14 at 18:10
  • yes!, but until @Noifma won't make us aware of all the possibilities ours, are just hypothesis :) – sirfoga Apr 29 '14 at 18:11
  • You are right that the second digit can't be $3$. You could comment that $42xx24xx$ doesn't work. This would complete the proof that your solution is correct. – Ross Millikan Apr 29 '14 at 18:17
  • @NotNotLogical and Foga thank you for your help. The question is indeed ambiguous but I could not ask the automated system for clarifications.

    In the end the correct answer is 41312432.

    That means that the correct format is 1-1, 2--2, 3---3, etc.

     – Noifma Apr 30 '14 at 19:44
  • I'm glad you've made us aware of any result at the end! thanks! ;) – sirfoga Apr 30 '14 at 19:46
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I assume by $1$'s are one space apart you mean the $1$'s must be adjacent. The best you can do is start with a $4$, giving $4bcd4fgh$. Now $b$ can't be $3$, so try $2$, giving $42c24fgh$. Clearly $c$ must be $3$ and we get $42324311$

Ross Millikan
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  • Thank you for your help. The question is indeed ambiguous but I could not ask the automated system for clarifications. In the end the correct answer is 41312432. That means that the correct format is 1-1, 2--2, 3---3, etc. – Noifma May 03 '14 at 11:30