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Durham 342 (Borthwick 88, Clark 86, Baker 6-46) beat Somerset 333 (Green 157, Rushworth 3-58, Gibson 3-54) by 11 runs
Durham posted 342 all out after losing the toss, Scott Borthwick making 88 from 67 balls, including 3 sixes and 11 fours, and Graham Clark a more measured 86. Nineteen-year-old seamer Baker claimed a career-best six for 46 from 10 overs as he tried to halt the one-way traffic.
Somerset plunged to 195 for eight in reply before Green - with little support - launched an incredible onslaught in the face of well-nigh impossible odds, crashing 12 sixes and 10 fours in an unforgettable inning of 157 off 84 balls.
It was only enough to boost the score to 333 all out, but provided rich entertainment for a crowd of 3,000 on one of the hottest days of the year.
Green said: "If only we could have found ten more runs out there. I have never struck the ball better and that's why I went for another six in the final over. I didn't quite connect right and the bottom line is that we have lost the game."
Earlier, Jonathan Bushnell, making his List A debut, contributed a composed 24 to a Durham opening stand of 60 in 13 overs with Clark before becoming Baker's first victim, miscuing a catch to fine leg.
Borthwick hit the first six of the match off Lewis Goldsworthy before Clark moved to a sold half-century, from 77 balls, with 6 fours.
Borthwick's fluent fifty occupied just 42 deliveries as Durham laid the foundation for a challenging score, reaching 124 for one at the halfway point of their innings.
Clark cleared the ropes off Green and Goldsworthy before falling with the total on 191, bowled having a swing at Green. He had faced 101 balls and hit 2 sixes and 10 fours.
Nic Maddinson kept the momentum going with 24 off 13 balls before being bowled attempting a big hit off Baker. The Somerset youngster removed Borthwick in the same over, the 36th of the innings, then pinned Sean Dixon lbw for 11.
The hosts had given a debut to 19-year-old left-arm seamer Alfie Ogborne, from Yeovil, who bowled four tidy overs for 21 and ran out Paul Coughlin with a throw to the bowler's end from deep mid-wicket to make it 260 for six.
Liam Trevaskis averted any thoughts of a Durham collapse with a rapid 44, including a six and 6 fours, while Tomas Mackintosh weighed in with 28 before driving a catch to long-on to give Baker his fifth wicket.
Oliver Gibson was yorked by Baker, whose previous best List A figures had been three for 46, and Durham's innings was wrapped up by another run out in the final over. Somerset's reply had reached 26 in the sixth over when Steve Davies' was caught behind for nine, chasing a wide delivery from Gibson.
Matt Renshaw survived a chance to Coughlin at mid-on off Chris Rushworth, but perished in the same over for 23, driving a catch to mid-off to make it 36 for two.
Playing in what may well prove the last competition of his illustrious Somerset career, James Hildreth was soon troubled by a muscle injury and made only 12 before being caught at deep cover off 18-year-old seamer Stanley McAlindon.
Goldsworthy (27) and George Bartlett (34) shared a stand of 56. But they fell in quick succession, Bartlett driving Gibson in the air to mid-off and Goldsworthy caught and bowled above his head by Borthwick to make it 120 for five.
Green and James Rew added 46 before the latter pulled a catch to deep square off Trevaskis. Kasey Aldridge fell leg before to Rushworth and Jack Brooks followed, but if Durham thought the job was done, Green had other ideas.
Having reached a 45-ball fifty, his maiden List A hundred occupied only 14 more deliveries as he blasted four sixes in an over off Borthwick and four more off successive balls from Gibson.
They were massive blows from the inspired captain, who continued to put Durham's bowlers to the sword in a frenetic finale, dominating stands with Ogborne and Baker.
Somerset needed 11 off the final over, sent down by Gibson. There was visible relief among the Durham players when the second ball saw Green well caught by Trevaskis diving forward at long-on.
Royal London One-Day Cup