I wonder if this would work out:
Turn 1: CCEEE: --> Lighthouse
Turn 2: CCCCC:--> Mint {reshuffle}
Turn 3: CCE, LH, Mint--> Play lighthouse, purchase Silver {reshuffle}
Turn 4: EESCM---> purchase remake
Turn 5: Remake, EE,LH,S---> play lighthouse, remake 2 estates to silvers, purchase another LH
Turn 6: SSEC, Mint---> Purchase Festival (or other +$2 card) {reshuffle}
Turn 7: SSC, LH, Remake--->Purchase Gold
Turn 8: Festival,Mint,S,C,E---> Play Festival, Purchase Gold {reshuffle}
Turn 9: LH,M,E,G,Festival ---> Play Festival, Play Lighthouse, Mint Gold, Purchase Gold
Turn 10: C, G,S,C,Remake---> Play Lighthouse, Purchase Province {reshuffle}
Turn 11: SSS,E,C---> Purchase Province
Turn 12: Festival, Mint, G,G,Prov. ---> Purchase Province {reshuffle}
Turn 13: GGC, Remake, Lighthouse ---> Play LH, Mint Gold, Purchase Province
Turn 14: GGC, Province, Lighthouse ---> Play LH, Purchase Province
Turn 15: Province, Province, GGC---> Purchase Province
Turn 16: Mint, S,S, Province, Estate, --> Purchase Estate (or Garden, or Island, or nothing) {reshuffle}
Turn 17: GG, Festival, Province, Province ---> Purchase Province
Turn 18: G,G,S, P, P--->Purchase 8th Province
OK, silly excercise, takes perfect shuffle luck, but all in all, had enough to buy a Province every turn from Turn 10-15, and 17 and 18.
I guess this just goes to show that Mint is a terrible opening.