Today is 21 July 2019, the 202nd day of the year.
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			Suppose that you have two gears as shown below, one fixed and the other free to rotate around the fixed one. Each gear has some number of teeth on it; let's call the number of teeth on the fixed gear n, and the number of teeth on the rotating gear m. Put a red dot on one of the teeth on the rotating gear, and a second dot by whatever slot on the fixed gear your dotted tooth happens to be resting in at the moment. Now let the free gear rotate around the fixed one however many times is necessary for the dots to come back into line again. Clearly, if m = n then the free gear will only have to go around once, but otherwise it may have to go around many times before the dots come back into line. Today's challenge: What single condition on m and n is both sufficient and necessary to ensure that the dotted tooth on the rotating gear visits every other slot on the fixed gear before coming back to rest in the dotted slot? Bonus challenge: Given any m and n, calculate the number of times the free gear must rotate around the fixed one before returning to the starting position. HINT Solution to yesterday's challenge THE MATH STORE LIVES! At last, the new Math Store is open, including all the downloads we have offered in the past. Please come poke around the shop and let us know what you think! NEW FOR 2012: Platonic Realms is at last being rewritten and moved to a new server. Please check back soon to see the new look, new content (together with all your old favorites), and an all new math store! This site is registered with the Internet Content Rating Association (ICRA), and is suitable for all ages. Because we value freedom of expression on the internet as well as the privacy and comfort of our visitors, Platonic Realms is also bug-free and blink-free. Platonic Realms uses cookies for site-traffic analysis, but these automatically expire soon after your visit. Our privacy policy is very simple: we ALWAYS ask our visitors before collecting any personal information, and we NEVER share information about our visitors with anyone else. Period.

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