List all the 3X3 arrays of nine distinct positive digits such that each of the rows and each of the columns is an increasing sequence.
Example: 1,2,5; 3,4,7; 6,8,9.

(In reply to How about...? by Ady TZIDON)

This isn't going to be easy to understand because I used a tree diagram and I'm not going to list every solution.  But I did solve analytically.

Letter the array

abd
ceg
fhi

then an array of digits can be given as a sequence of these 9 letters.

clearly a=1 and i=9

either b=2 or c=2 (we need only consider b=2 because c=2 will give the same solutions by symmetry.  At the end, double the count)

so either d=3 or c=3 

The first of these forces c=4 then starts abdc there are 5 solutions from there.

The second allows d=4, e=4 or f=4

From here there are 5,6,5 solutions respectively.

For a total of 5+5+6+5=21

double this for c=2 and the grand total is 42.

Posted by Jer on 2015-12-02 10:54:33