# Weekly Brain Teaser | Consider a vertical wheel of radius 10 cm

Every week we feature a Brain Teaser or a thought-provoking question.

If you have one that you would like to share with fellow engineers, please email it to info@engineeringdaily.net.

So, here is this week’s,

Consider a vertical wheel of radius 10 cm. Now suppose a smaller wheel of radius 2 cm, is made to roll around the larger wheel in the same vertical plane while the larger wheel remains fixed.

**What is the total number of rotations the smaller wheel makes when its center makes one complete rotation about the larger wheel?**

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Here is the magic trick to this puzzle no one has pointed out yet:

Imagine you open up the large wheel into a straight line. Easy math tells us the small wheel will roll 5 times around if it travels the length of the straight line. Well if the line is curved back up into a circle (wheel), then the line is the same length and the small wheel has not changed, so why does it take 6 (instead of 5) rotations to go around? The answer if 5 + 1 really. The 5 rotations along the line PLUS one more big rotation of the small wheel around the big wheel.

Another way to look at is assume the big wheel was a single point. The outer wheel would not “roll” one bit. A mark on the outer wheel where it touches single point would stay put. Yet the outer wheel still rotates 1 time around the point. 0 + 1 times.

This works for any two wheels. Do the simple math like the inner wheel is a straight line and then add 1 rotation. This interesting single extra rotation does not change no matter the size of either wheel. It is always one more rotation.