Wikipedia:Reference desk/Archives/Mathematics/2008 September 29

= September 29 =

Is this true?
I read this on a website. I have a feeling that it should be obvious, but it isn't to me:


 * Three wheels. The first moves a any positive rate. The second moves at any positive rate different than the first. The third moves at Pi the difference of the two. If all three wheels stated at the same point, they never realign.

Is it true? If so is it something to do with Pi not being representable as a decimal or fraction? -- Q Chris (talk) 08:40, 29 September 2008 (UTC)


 * If I understand the question, you have three angles that vary with time: A, B, and C, where B = D*A for D>1, and C = pi*(B-A) = pi*A*(D-1). You want to know whether A, B, and C will ever all coincide, assuming they all start at zero and that they spin forever. In that case, yes, they might, but they won't be in their original position when they do. Replace pi with any irrational number E. (I'm going to measure angles as fractions of a whole angle here.) As they spin, two discs coincide when the difference between their angles is an integer multiple of a whole angle. So A, B, and C coincide when A-B = F and A-C = G for integers F and G. So A-D*A = F and A-E*A*(D-1) = G. So A = F/(1-D), so F(1-E*D+E) = G(1-D). Solving for D, we get D = (G-F-E*F)/(G-E*F) = 1+1/(E-G/F). That means that for each choice of E, F, and G, there is a single speed D which will cause all the discs to coincide at that time, and the set of all such values of D covers the real numbers densely. For a given E, such as pi, that means there are infinitely many speeds D for which the discs coincide eventually, within any possible range of values for D. On the other hand, say they all have to come back around to the beginning at the same time. Then A, B, and C are all integers, and E = C/(B-A), making E rational. Black Carrot (talk) 09:47, 29 September 2008 (UTC)
 * Thanks for your answer, but I think I have confused myself further. I thought I had proved the opposite, i.e. that they would never coencide!
 * In trying to understand the problem myself I though of the special case where A = 0, B=1 and C=pi. In other words you have a fixed point and two wheels rotating where the B moves at a rate of 1 and C moves at a rate of pi. In this case wherever b comes round again the angle C will be pi times B. Since B will have turned integral number of turns T. C will only also be opposite A if C * T is integral, so B * T = (some integer) * C * T
 * so B = (some integer) times C
 * This clearly can't be the case since B/C is irrational, which contradicts your answer. I had then thought that I could generalise this special case by considering the whole three wheels to be on a further wheel, rotating at the same speed as A but in the opposite direction! Help, I'm confused! -- Q Chris (talk) 10:14, 29 September 2008 (UTC)
 * For that special case, you are correct, but it isn't really a valid case since zero isn't positive (it's non-negative, but that isn't the requirement). Consider this case: The first moves at pi, the 2nd at pi+1, the third at pi, all three will coincide after 2pi units of time (one counterexample is all that's required to prove something isn't true). For it to be impossible you would need to introduce some restrictions such as the first two speeds being rational and the amount of time being rational, then it is easy to show it's impossible. --Tango (talk) 14:22, 29 September 2008 (UTC)
 * I have been thinking about the above, and I am still can't see it. Is your above example correct? After 2.pi units of time
 * A will have moved 2pi.pi units
 * B will have moved 2pi(pi + 1) = 2.pi.pi + pi units
 * C is the same as A
 * I don't see how that is the same. I am sure I am missing something obvious, but I don't know what. -- Q Chris (talk) 07:13, 30 September 2008 (UTC)
 * You made a mistake in B: 2&middot;pi&middot;(pi + 1) = 2&middot;pi&middot;pi + 2&middot;pi, which is exactly 2&middot;pi (i.e. exactly one full turn) more than A and C. --CiaPan (talk) 08:04, 30 September 2008 (UTC)
 * Thanks, so I did! Great I understand it now. So what this boils down to is that for some ratios (rational ratios) the wheels will never align again, but there are some ratios where they will. Is this right? -- Q Chris (talk) 08:20, 30 September 2008 (UTC)


 * Yes. To find infinitely many such ratios, plug random numbers into the formula I gave. Black Carrot (talk) 22:39, 30 September 2008 (UTC)


 * For a related problem, see Lonely runner conjecture. Robinh (talk) 07:01, 1 October 2008 (UTC)