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Very clever and difficult number theory puzzle (with generalization) 
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#1
Jun2808, 04:25 PM

P: 25

For any 10 digit natural number [tex]N[/tex] in which
the first digit corresponds to the total no of 1's. the 2nd digit corresponds to the total no of 2's. . . . the 10th digit corresponds to the total no of 0's. determine, with proof, if the number of such natural number [tex]N[/tex] is finite, and if proved true, find them all. A generalization of http://answers.yahoo.com/question/in...8051813AA0p296 Also, extend this to any numerical base [tex]M[/tex] such that the [tex]M^{th}[/tex] digit corresponds to the total number of 0's and [tex](M  1)^{th}[/tex] digit corresponds to the total number of [tex](M  1)[/tex]'s for any natural number [tex]M[/tex], etc. 


#2
Jun2908, 04:38 AM

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The first part seems quite trivial, as the number of 10 digit natural numbers is finite, so certainly the number of such special 10 digit natural numbers is finite. The question is, how to construct them all? And are numbers like "0100000000" a valid example (because technically, this is a nine digit number).



#3
Jun2908, 09:20 AM

P: 688

I suspect the constraints are so tight that no combination will succeed (find a working example!).
This is the case for base 2, where none of the 4 possible combinations 00, 01, 10 or 11 match the requirements. P.S.: Also, since the sum of digits must be exactly M, and all individual digits must be <M, there remain few cases to verify for bases 3 and 4, being readily visible that those bases do not work either. 


#4
Jun2908, 10:30 AM

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Very clever and difficult number theory puzzle (with generalization)
And I take back the example I gave, as there should be a number in the last slot to indicate the number of zero's. In fact, I can't think of an example at all at the moment.



#5
Jun2908, 10:31 AM

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#6
Jun2908, 10:40 AM

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#7
Jun3008, 07:17 AM

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There are one or two solutions for base 4, depending on whether 0202 counts as a solution. These are the only solutions up to but not including base 7. There is a general solution for base b=7 and up: b4 zeros, one b4, one 2, and two 1s: 2110003, 21010004, 210010005, 2100010006, etc. Without proof, I posit that this general solution is the only solution for base 7 and up.



#8
Jun2809, 09:04 PM

P: 1

2100010006 b10
1000000008 b10 8111111110 b10 These came from the Yahoo! thread... 1o57 


#9
Jun2809, 09:43 PM

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8111111110 is not a solution. This number has one 0 in it, but has a zero as the tenth digit. 2100010006 is a solution. 


#10
Jun2909, 02:12 AM

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