Of course, the variety matters.
This is called 'Linda' and they don't get very big, even under best conditions.
As for pattern of planting: I think as long as they have enough space, the pattern doesn't matter all too much.
That being said, for my circumstances here, the most important factor is rain;
If I get no rain, they are chicken-egg sized, with good rain, they are about the size of a human fist.
You're right, rain is one of the most important factors. Tuber crops have a way of pulling some good surprises though. I hope by September it'll be a good harvest