How many stopped clocks do we need to always have the right time? Imagine you have a finite collection of 12-hour clocks which we can set but none of them work. How many do we need so that at least one of them has the right hour? It should be obvious that the answer to this is 12, one for each hour. Now imagine that were in some crazy alternate reality where clocks have all different sorts of numbers of hours on them. Our collection is particularly weird as NONE of our clocks have the same number of hours on them. Can we still set them so that, no matter what the hour is, one of them is right? To give an example: suppose we have a 2-hour clock set to 0, a 3-hour clock set to 1 and a 5-hour clock set to 2. Then 27 hours after we start, the 5-hour clock will be right but weve not covered every time as 3 hours after we start, none of them will be right. Bob Hough proved last year that we can always do this, provided that the clock with the fewest number of hours on isnt too big. The best estimate at the moment is that there has to be some clock with fewer than 1,000,000,000,000,000,000 hours on it! This answered a long-standing conjecture of Pal Erdős in the negative. This is an expository talk by Ben Green on Houghs proof.
Posted on: Wed, 17 Sep 2014 20:36:41 +0000
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