ANSWERS: 2
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Whether the batsman offers or not, as long as it hits the bat then hits the boundary then the four points are scored. If it doesn't hit the bat then no points are scored. My mistake, I thought I knew the answer but after researching it further I accept that "The Knife's" answer is in fact correct.
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In the instance described then 4 byes should be awarded. This often happens when a batsman ducks underneath a bouncer which then passes over the wicketkeeper's head, for example. The rule for leg byes is different as you seem to be aware - if the ball were to brush the batsman's pads or body and run away to the boundary without him offering a shot then a dead ball would be awarded, this is to prevent batsmen from deliberately deflecting the ball for runs with their body. But if no contact is made with any part of the batsman and the wicketkeeper isn't good enough to stop it, that isn't the batsman's problem and so the 4 runs will stand as byes.
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