Making and doing things infinitely can get a bit tricky, but we’ll give it a go. Find yourself a box and a lot of balls and number the balls, starting 1, 2, 3, 4 . . . You get the picture. The game is to put the balls in the box one at a time, starting with ball number 1, but, whenever you put in a square-number ball, you take out its square root and put it away in a drawer or somewhere safe. This means that the first move is a bit odd, because when you put in ball 1, 1 is its own square root, so you take it straight back out. Then you can put in balls 2 and 3. When ball 4 goes in, you take ball 2 back out and put it in the drawer. Then balls 5 through to 8 go in, before adding ball 9 and taking out ball 3. The question is: if you simply keep doing this, which balls will end up in the box? How many are safe in the drawer?
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