Testing our 90-led charlie-plexed LED array

There's something not quite right with our LED array.
In the main, it works fine. By simply applying 5V across different terminals we can see individual LEDs light up. Here's an example of lighting a single LED by applying 5v across two of the 10 available terminals:

(yes, that is Jason's traffic light LED project - I just nicked the 9v battery and 5v regulated output for the sake of testing this new board!)

But every now and again - and it is just every now and again, not very often - we get a peculiar result. Instead of single, bright, LED lighting up, an entire row (and sometimes part of the column) lights up too. We're putting this down to either a dodgy LED somewhere, or an LED being soldered in the wrong way around.

At the minute there's no real pattern to which combination of terminals gives the bad response.
There's one way to get to the bottom of this - to build an LED tester:
The idea is to create something that we can control which pin takes 5v and which takes ground, and record the result in a truth table (similar to the table created at design stage). Hopefully by cross-referencing these results to our schematic, we should be able to identify which LED(s) are common to all the problem terminals.