By Scott Kim|Wednesday, August 01, 2001

Circular Reasoning

All you need is a nimble mind (a few quarters and a really, really big ball of string might also help) to get your head around these well-rounded stumpers

Rolling Around

1. [Easy] Two quarters sit on a table so that both portraits face left and the bottom of one coin touches the top of the other. If you put your thumb on the lower coin to hold it in place and allow the other to roll counterclockwise around it without slipping, as if the edges of the coins were interlocking gears, in what direction would the portrait face when the upper quarter reaches the bottom?

2. [Medium] Which way does the rolling quarter face when it reaches the bottom if the diameter of the stationary quarter is twice that of the rolling quarter? What if the diameter of the stationary quarter is exactly half that of the rolling quarter?

3. [Medium] Now picture two world globes stacked one atop the other. Both globes are oriented the same way, with the north poles pointing up. Hold the lower globe still and roll the other around it without slipping or twisting. When the rolling globe reaches the bottom, which of its points will be touching the south pole of the stationary globe?

4. [Hard] What is the answer to problem 3 if the diameter of the stationary globe is twice that of the rolling globe? Again, assume the globe rolls without slipping or twisting.

Round Trip

1. [Easy] An explorer somewhere on the Earth walks 10 miles south, 10 miles east, and 10 miles north and yet ends up at the same point where she began. Where could the explorer have started to allow that result? Is there a second answer to this problem?

2. [Medium] An explorer somewhere on Earth turns to face south and walks forward 10 miles, then walks east 10 miles, turns to face south again and walks forward 10 miles, and finally walks west 10 miles. At the end of her travels, the explorer ends up right where she began. What was her starting point?

3. [Tricky] An explorer turns to face south and walks forward 10 miles, ending at her beginning point. Where did she begin?

Around The World

Suppose Earth is a perfect sphere 25,000 miles in circumference, and you are required to tie a massive loop of string around the equator. Due to a slight miscalculation, your loop of string is 20 feet too long. You start thinking about ways that you can pull the string taut.

1. [Easy] How far off the ground would you need to suspend the string all the way around the equator in order to take up the slack? Hint: The suspended string is a perfect circle, concentric with Earth.

2. [Medium] If you wanted to pull the string taut by placing a pole under the string at one point on the equator, how tall a pole would you need? Hint: You'll need a bit of trigonometry to solve this one.


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