Puzzles/Div9
Find a 9-digit number , containing all the digits 1,2,...,9, such that divides for each .
Solution
Since 5 divides , must be 5. Similarly must be even for even and by elimination , , and must be the remaining digits 1, 3, 7 and 9 in some order. Candidate solutions thus have the form oeoe5eoeo where o=odd, e=even.
From this point it's not too difficult to reduce the candidates to a unique solution; here is how one might use J:
oe=: (i.24)&A.&> 1 3 7 9; 2 4 6 8 NB. permuted odds and evens a0=: 5 ,. ,/ ,"1// oe NB. proto-candidates (5ooooeeee) a =: 1 5 2 6 0 7 3 8 4 {"1 a0 NB. candidates (oeoe5eoeo) ok=: 0 = (# | 10&#.)\"1 a NB. 0 remainders in all prefixes (*./"1 ok) # a NB. solution 3 8 1 6 5 4 7 2 9
Still using the same oeoe5eoeo idea, but building the solution one column at time:
nd =: 0 1 0 1 2 1 0 1 0 { 1 3 7 9 ; 2 4 6 8 ; ,5 ext =: [: ,/ ] ,."2 0 {:@$ {:: nd"_ ok =: *./@~:"1 *. 0: = {:@$ | 10&#. extend=: (ok # ]) @ ext extend^:9 i.1 0 3 8 1 6 5 4 7 2 9
Initially, there is 1 solution of 0 columns. i{::nd are the digits to use when the current solution has i columns, and ext appends each of these digits to each row of the current solution. ok indicates the valid rows -- unique digits and divisible by i+1 .
The method keeps the number of rows small, and therefore might be the method to use if one were to solve the problem by hand (with only pencil and paper).
3 : '# extend^:y i.1 0'"0 i.10 1 4 16 20 30 30 20 8 1 1
Contributed by Roger Hui and Ewart Shaw.