Why can't someone build a custom computer and win the reward for a few days? I've been reading about mining, and that to get the reward it basically comes down to brute force. The difficulty of this brute force being adjusted every 2016 blocks.

I also understand that some hardware are better at brute force than others.

So, if I were to tinker in my garage an amazing mining rig, connecting together many ALUs or whatever (sorry I'm not a hardware guy) to give myself a significant advantage - would this work?

I'm not saying it would work forever, but would it work until someone built an even better machine, or until the difficulty self adjusts again?

I assume it's not the case since it would happen a lot. But seeing how people are racing to build the top mining rig, I feel like there is some degree of truth?
Lets see, following on from the other answer, you're going to need 6,400,000 TH/s. Currently you can buy the Antminer S9 which will give you about 13.5 TH/s, so you'll need a mere 475000 of them.

If you are buying  that many units I reckon you could negotiate a pretty good discount, if you don't cause an international shortage. Lets say you could get them for $3000 a unit. This would cost you only $1,420,000,000.

You are going to need about 3900 m/3 of space to store the devices, unpacked. By my reckoning that is about 4 tennis courts filled about 2 meters deep. Not a vast area but bigger than most garages. In reality, you'll need more space for ventilation and access.

To run all the devices concurrently you'll need about 66,500,000 Watts of power, that is enough to supply about 88,000 domestic homes. Your electricity bill would be at least $50,000 a day.

By my reckoning, you'll need about 45,000,000 BTU's of cooling, this could be achieved with 661 of these costing around $7,000,000. I haven't accounted for the additional space, electricity and airflow requirements.

The S9's internet bandwidth requirements are relatively economic, averaging out at something like 500 bytes a second. So, you'll only need 238 Mb/s bandwidth. This could be be supplied via and OC-12 line which will cost at least $700 a day, or more depending on distance. You're going to need something like a 1 Cisco Catalyst 6513-E switch, 180 Cisco Nexus 2348TQ and 10,000 Cisco Nexus 2248TP-E switches. The switches alone will cost something like $51,000,000. There is likely a more cost effective solution but you'll need to speak to your engineering team to work that out.

None of this takes in to account other concerns like expert cable management and racking, maintenance and security.

Finger in the air, you need a couple of billion dollars to get started. Given the current market cap this might actually pay off but you can see the risk and, don't forget the cleanup cost once you have devalued the market. 

Good luck.