20. It turns out there is a final exam for this course. I give you 8 marbles that are exactly alike except that one of them is slightly heavier than the others. I then give you a balance scale and ask you to determine the heavier marble in 3 weighings. Will you pass your final exam?

21. Okay, you got through that one, but now it turns out that you don’t obtain math certification unless you can determine the heavier marble in only 2 weighings. Will you get your certification?

22. Why does time seem to pass more quickly as we get older? (I’m not just making small-talk here; there’s an actual mathematical reason for this. And I’m not suggesting that any of us are getting old—although you may feel that you’ve aged just because of these quantitative reasoning exercises.)

24. You and a friend stand side by side and begin walking so that your right feet hit the ground at the same time. You take exactly 3 steps for each 2 steps of your friend. How many steps go by before you and your friend’s left feet simultaneously hit the ground?

26. Enough fairy tales; let’s do something down-to-earth. Suppose the earth is a smooth, perfectly round sphere. Imagine a rope pulled snug around the earth’s equator. If we add 10 feet of length to this rope and then prop it up evenly all the way around the planet, how far above the earth’s surface would it be? And since this exercise may feel a little more like math than the rest, here are 4 answers to choose from:

a) less than one millimeter

b) between one millimeter and one inch

c) between one inch and one foot

d) greater than one foot

I've done the others already. These are just what I need help on still.

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