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 20151202 10:54:33 