No chance of getting a through bus. The train options are limited and go via England. By car can take forever.
There is however, according to Google Maps, an air option:
http://goo.gl/iEzmG9Hah, funny! Not sure a dragon can constitute public transport though.
Jolly good! I see that Fort Augustus to Urquhart Castle gives an equally appropriate option...
That is slower than the road though. It's quicker by dragon...
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I know this is posted in 'The Lighter Side', but for the benifit of anyone looking at this expecting a discussion on public transport between north and south Wales then there is in fact a real-life air link (Anglesey to Cardiff though, not Snowdon to Brecon), which I don't agree with because it represents an official endorsement of one of the least sustatinable modes of transport. If you want to be 'green', the best way of going from Brecon to Snowdon is probably to catch the T(C)4 TrawsCymru service to Newtown, then a train to Porthmadog (at the moment it will be rail replacement coach/bus north of Harlech). Not sure if there is a direct bus from there to the foot of Snowdon, but if not you can get one to Caernarfon where there probably will be buses to the foot of Snowdon. Or you could use the Welsh Highland Railway from Porthmadog to Rhyd-Ddu and start your climb there.