I was basically going to go the ABSOLUTE cheapest route, and that is basically screwing the motherboard to a piece of wood or making a cheap case and putting a cheap CPU, maybe 1 GB ram, then a couple mid range video cards, psu, and call it a day. I would install Ubuntu via USB external drive and use USB wireless mouse keyboard.
*This is all hypothetical right now, so ignore the case/wood, and any cost of power, etc.
1. Do i need more than 1GB to improve performance (since this machine would be for mining only).
2. Will the CPU make a difference?