I became stuck on this puzzle I found in a newspaper and assumed it was do-able but didn't know by which logic an expert would use to solve it. Can someone explain where I should have gone from this point:

2 7 * * * * * 4 8

1 * * * * * 2 7 3

* * * 7 * 2 * 9 *

9 * 5 1 7 8 4 * 2

8 * 7 4 2 6 9 * *

* * 2 5 9 3 8 * 7

7 * * 2 * 4 * * *

5 2 * * * * * * 4

3 8 4 6 * * * 2 9

I knew column 1 still needed a 4 & 6 so I put a 4 in r3c1 and solved. What technique should I have used to prove that this was right?