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June 17th, 2012, 06:54 AM  #1 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Rearrangement of 123456789
K=123456789 is a nine digits number now we rearrange randomly the digit in K and will get 9 ! different numbers if abcdefghi is one of this number for the sum of any adjacent three digits (example :a+b+c ,b+c+d . c+d+e,.............g+h+i) (i) please find the maximum value of N Ans: max(N)=15 (ii) How many different numbers we can find satisfying the rule of (i) Ans:124 
June 24th, 2012, 04:47 AM  #2 
Member Joined: Jun 2012 From: UK Posts: 39 Thanks: 0  Re: Rearrangement of 123456789
This is not a full solution, and I don't intend to add anything to it, and there may be better ways. Firstly there is no maximum value for N. There is however a minimum. Here's some opening thoughts: If we add all 7 equations, then we have a+2b+3(c+d+e+f+g)+2h+i <= 7N The minimum value of the LHS arises if we assign 9,8 to a and i, 7,6 to b and h, and the rest to c,d,e,f,g. Doing this we end up with 88 <=7N, and hence 13<=N, since N must be an integer. So, 13 is a lower bound for our N. Can we construct a number where N is 13? Well no. Why not? If we consider the digits 9,8,7, then none of them can be in the same group of 3 adjacent digits as each other (as they would give a higher value of N). So: If we divide our 9 digit number up into 3 lots of 3 (i.e. abcdefghi), then each of the digits 9,8,7 must be in a different cell. We now need to allocate 1,2,3,4,5,6 so that the maximum value of those cells is as small as possible. It turns out that this is 15, and this our new lower bound for N. The question now is, can we construct a number where N is 15? And the answer is, yes.... 
June 25th, 2012, 12:07 AM  #3 
Member Joined: Jun 2012 From: UK Posts: 39 Thanks: 0  Re: Rearrangement of 123456789
Contrary to my previous opening sentence, here is a bit more, since ~99% of what I wrote previously is redundant. If we consider the sum of the 1st three digits, and the sum of the 2nd three, and the sum of the third three, then each of these sums must be less than or equal to N, and their total less than or equal to 3N. But the total is just the sum of the digits 1 to 9, and equals 45. So 45 <= 3N and thus 15<=N. It just remains to construct numbers where N=15, to show min(N)=15. 
June 25th, 2012, 03:27 AM  #4  
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Re: Rearrangement of 123456789 Quote:
for the sum of any adjacent three digits must <=15 so it shouls be max(N)=15 ,not Min(N)=15 for example : (384 267 159) is only one situation of all rearrangements: 3+8+4=15, 8+4+2=14, 4+2+6=12, 2+6+7=15 , 6+7+1=14 , 7+1+5=13 , 1+5+9=15 (here 15 is max(N) ) How to get the second answer (124),this part is more difficult ,it must be something related to permutation I wrote a computer program to get the answer (124)  
June 25th, 2012, 03:36 AM  #5  
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Re: Rearrangement of 123456789 Quote:
Quote:
 
June 25th, 2012, 04:00 AM  #6  
Member Joined: Jun 2012 From: UK Posts: 39 Thanks: 0  Re: Rearrangement of 123456789 Quote:
Since you're choosing any number you could choose 123456789 and here the max of 3 adjacent numbers is 24. If all sums are less then something, then they are less than any number bigger than that something, so there is no max. to that something. Quote:
 
June 25th, 2012, 06:55 AM  #7 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Rearrangement of 123456789
If, as you say, the digits in K can be rearranged randomly to produce any of the 9! expressible by these 9 digits, then the sum of any three consecutive digits can range from 1+2+3=6 to 7+8+9=24. So in what sense would 6 not be the minimum and 24 the maximum? Of course, 6 and 24 won't appear in all arrangements. So you can ask minimax and maximin questions, ie what it the number n such that the largest sum of any three consecutive integers is never smaller than n and what is the number m such that the smallest sum of any three consecutive integers is never larger than m. Is this what we are talking about here? 
June 25th, 2012, 06:59 AM  #8 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Re: Rearrangement of 123456789
If you let N<15 and again test it with your program ,you will find no answer, that is why I set max(N)=15. May be I did not make a clear definition 
June 25th, 2012, 07:34 AM  #9 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Rearrangement of 123456789
Can you show an example of an arrangement where the highest sum of three consecutive digits equals 15? I'm pretty sure that the highest sum will never be lower than 16 and the lowest sum will never be higher than 14.

June 25th, 2012, 07:55 AM  #10 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Re: Rearrangement of 123456789
(384 267 159) is only one situation of all rearrangements: 3+8+4=15, 8+4+2=14, 4+2+6=12, 2+6+7=15 , 6+7+1=14 , 7+1+5=13 , 1+5+9=15 (here 15 is max(N) ) 

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