Precedent and probability are set against England. Only once has a country chased down a target of more than 315 at Headingley, and that was not England but Australia, and they needed the services of a chap called Bradman who, as normal, scored a voraciously large hundred.
New Zealand still have five overs to go before England can take the -second new ball and they are already 338 runs ahead. Barrie-Jon Watling, dapper as only a keeper can be, has become the first New Zealander to score a century at Headingley and, with four wickets still left, is liable to set England something near 400.
It is more than theoretically possible that England can win, which would considerably impress the Australians. Two of their batsmen would have to make a hundred, but nine of them have done so in Tests. It had come to seem ever more implausible that Stuart Broad had ever made one, but in smacking 46 and taking England to parity with New Zealand on first innings, he batted his best since his nose was broken last year by India's Varun Aaron.
England will also have the best part of two days to chase their target down, so the run-rate required will not be daunting if rain keeps away. They can take their time, if not quite so much as when they accomplished their highest run-chase of 332 in 1928?29 against Australia in a timeless Test.
New Zealand's desire, however, has been more intense throughout this game. It has been manifest in the stricter discipline maintained by their pace bowlers, who have not pursued exotic options. It has been manifest in the greater hunger of their batsmen for quick singles, the means by which New Zealand activated their second innings after the loss of two early wickets, before their attacking strokeplay took the match away from England.
New Zealand, in addition, will have access to two new balls as they pursue what would be only their fifth Test victory in England. It has been a batting pitch - supplemented by a fast outfield - except when the ball has been new, as it was when Tim Southee took the lead in reducing England from 215 for one to 267 for eight before Broad's assault.
Five one-day matches and a Twenty20 international will follow this Test, but it might be more entertaining if a third Test was staged instead - and New Zealand's run-rate in Tests is now barely less than it is in ODIs. An ideal venue for this third Test would be Alton Towers, as this series has been one prolonged switchback, with never a dull over let alone a dull session.
The third morning, on another cold and windy if sunny day, was yet another big dipper. When England resumed at 253 for five, Southee dismissed Jos Buttler, Ian Bell and Moeen Ali with perfectly pitched balls which brought catches at first, second and third slip.
Fortunately for England, the newish ball was then dampened by the residues of the overnight rain, and Broad began to swing from the hip. His stand of 51 from 49 balls with the more orthodox Mark Wood was the second highest of the innings, and for the last two wickets Broad engineered 83 off 89 balls.
England's innings resembled a sandwich made of crusty bread with a lick of butter in between.
Relieved to have overcome this psychological hurdle before Mitchell Johnson hits town, Broad found his rhythm from the rugby stand end to flush out Tom Latham and Kane Williamson. The wind had veered to blow from the west, so Broad was no longer running into it as he had on the first day, when he declined to bowl more than three overs in his opening spell after New Zealand had been sent in.
Thereafter, beginning with singles pushed square on the off side, New Zealand counterattacked with their now customary gusto. From being the most defensive Test team a -generation ago, they have become the most attacking: so long as Chris Rogers keeps opening for Australia, New Zealand can claim that title.
Martin Guptill's long levers -disposed of any width, and there was enough width for him to hit 70 off 72 balls. With Ross Taylor scoring 48 at a run a ball, New Zealand's third-wicket partnership of 99 off only 87 balls may well turn out to have been the match?winning one.
Another area in which there has been a distinct difference between the teams, in addition to bowling discipline and batting hunger, has been the treatment of off-spin.
Mark Craig was allowed to bowl his first 23 overs for only 38 runs as Gary Ballance scored a single off the 15 balls he faced from Craig; Cook was as passive as he had been against Marlon Samuels in the Barbados Test, scoring 13 off 63 balls. Thereby England's first-innings score was not close to New Zealand's total when the danger point of the second new ball arrived on Saturday evening at 236 for two.
New Zealand's batsmen went the whole hog against England's off-spin. Craig has bowled 26 overs for 48 runs on the same pitch that Moeen Ali and Joe Root have bowled 31 wicketless overs for 141 runs, although Moeen has had three chances missed. Watling, with his slog-sweep, did his share in never allowing Moeen to settle. He even dominated the stand of 121 with Brendon McCullum, who made his highest score of the series, before receiving a decision from -Sundaram Ravi that was correct but not a good one.
England's pace bowling was not so wild as it had been first time round, and Wood maintained his progress, but while he reversed the swing he could not reverse the tide.