Peter often passed away time skimming pebbles across the nearby river above the weir, where the water was still and deep. From his observations. Peter reckoned that each bounce of the pebble was half the length of the previous one. He found he could easily skim a pebble the full width of the river whenever the first bounce landed more than halfway across the river. However, he was determined to succeed when the first bounce landed exactly halfway across, and persisted in trying. If Peter succeeds in skimming a pebble so that its first bounce does land exactly halfway across, how many bounces will it take for the pebble to reach the other side of the river?

A conjuror arranged before his audience a line of seven identical boxes, in which were a number of coloured balls. He invited his audience to give him any number N from 1 to 24, and he would then be able to tip out either a single box containing N balls or an adjacent set of boxes which between them contained N balls.
How Many balls did each box contain?
It may help you to know that no number N from 1 to 24 required more than five boxes to be emptied!

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