Comments on: Weekly Brain Teaser | Consider a vertical wheel of radius 10 cm

By: Mike Sprayberry

Mike Sprayberry — Fri, 10 Jul 2009 17:16:50 +0000

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.

By: admin

admin — Mon, 15 Jun 2009 15:16:31 +0000

Thank you Kevin.

By: Dave

Dave — Tue, 09 Jun 2009 11:41:54 +0000

You’re right Kevin! Six rotations is correct. Your drawing showed that a point on the smaller wheel would sweep 432 degrees of arc before contacting the same point again with respect to the larger wheel. Interesting to note that a 2 cm radius circle would make five 360 degree rotations when traveling from start to end of a straight line that is 2π x 10cm long.

By: Artie

Artie — Tue, 09 Jun 2009 10:33:55 +0000

Yes – 5 rotations.

By: Dave

Dave — Tue, 09 Jun 2009 10:32:20 +0000

Nice drawings Kevin. Try the cardboard cutouts and you will find the answer is 5 rotations.

By: KevinC

KevinC — Mon, 08 Jun 2009 21:46:25 +0000

and yeah i used 10cm and 2cm as diameters instead of radius but its still proportional to the original problem

By: KevinC

KevinC — Mon, 08 Jun 2009 21:41:53 +0000

http://img504.imageshack.us/img504/959/math.jpg

i drew this in ms paint. are you still sure its 5 rotations?
imagine the large wheel is stationary and the smaller wheel is rolling along its edge.
the smaller wheel is marked to indicate how much it rotates.

i believe the first example is what most people think is the correct answer, but it isnt.

By: Artie

Artie — Mon, 08 Jun 2009 14:41:35 +0000

Yes.

By: KevinC

KevinC — Mon, 08 Jun 2009 05:18:22 +0000

artie are you sure your “rotations” are 360 degrees?

By: admin

admin — Thu, 04 Jun 2009 16:53:13 +0000

It appears that we can’t seem to agree on the correct answer. Can anyone provide a video, schematic excel worksheet or other type of evidence to back their answer?
Thank you all for participating.

By: Artie

Artie — Wed, 03 Jun 2009 18:06:28 +0000

I kept the 10 cm radius circle stationary and rotated the 2 cm radius circle about it, marking each complete revolution on the larger circle. After one trip around the large circle there were 5 equally spaced marks, not six. Good call Dave.

By: KevinC

KevinC — Wed, 03 Jun 2009 01:40:00 +0000

also both wheels are on the same 2-dimensional plane

By: KevinC

KevinC — Wed, 03 Jun 2009 01:38:37 +0000

you have to remember that the large wheel is stationary and the smaller wheel is moving along the edge. thats the key to this brain teaser.

it takes 6 full rotations (360 degrees each) to make it all the way around

if both wheels were stationary, like a couple of gears, then it would be 5 rotations

By: Artie

Artie — Mon, 01 Jun 2009 16:21:58 +0000

5 is correct – I gave it a spin.

By: KevinC

KevinC — Mon, 01 Jun 2009 08:23:31 +0000

6 is the correct answer.

If you can not visualize it, make yourself some cardboard cut-outs and give it a spin. Empirical evidence will always trump endless babbling.

fixed