10 digit number puzzle

[MattcAnderson]

The only solution I know about is 6210001000.
Luis, Science man and Mawn, thank you for participating.
I found this puzzle at www.mathisfun.com

Regards,
Matt

[Mini-Geek]

Originally Posted by TheMawn

And sure enough:

B21,000,000,001,000
A2,100,000,001,000
9,210,000,001,000
821,000,001,000
72,100,001,000

Essentially, there's as many 0's as you want, a first digit indicating that number of zeros, and then a 1 for that digit. Then, you trigger an extra 1 for the number of 1's, giving you a 2 for the number of 1's, giving you a 1 for the number of 2's (which conveniently replaces the 1 for the number of 1's you had earlier.

Numbers like the one solving this puzzle, but for more generalized bases, were discussed at Math SE.

[science_man_88]

Originally Posted by retina

This is an old puzzle. And since the first two posters haven't yet found a result I'll not give away a solution edit: I see one poster has since edited and given a solution for base 10.

But there is a simple function that can generate such numbers for any arbitrary integer base except 2, 3, 6 and 7. I think it would be more interesting to try and find the general function first and then get the answer for the OPs question in base 10.

tendigitpuzzle(p,a)=for(x=a^(p-1),a^p-1,if(vecsum(digits(x,a))==p,b=0;for(y=0,p-1,if(#select(v->v==y,digits(x,a))==digits(x,a)[y+1],b=b+1));if(b==p,print(x))))

I actually solved it base 10 first. and yes I know PARI is slow. edit:okay there's a much easier version of this in theory using partitions that are padded with 0 and can't include the number of digits minus 3,2,or 1 at least instead of changing it to print digits(x,a).