You asked about the "cheapest possible way to mine".
If you're on a tight budget, don't want to wait for hard-to-find hardware, you could always do some hashing by hand using pencil and paper.
see: http://www.righto.com/2014/09/mining-bitcoin-with-pencil-and-paper.html
Cool link! But I doubt it'd be profitable considering the amount of pencils and paper you would use If you're on a tight budget, don't want to wait for hard-to-find hardware, you could always do some hashing by hand using pencil and paper.
see: http://www.righto.com/2014/09/mining-bitcoin-with-pencil-and-paper.html
On topic: Check out some CPU-only coins. You wont need to buy any new hardware to mine those and laptops are quite power efficient.