I recently found out about this interesting problem:
https://math.stackexchange.com/questions/1840996/watches-on-a-table#=
- You have N watches on a table, all in different orientations, each watch might be of a different size, prove that there exists a moment in time when the sum of the distances from the center of the table to the ends of the minute hands is more than the sum of the distances from the center of the table to the centers of the watches.
- How should each watch be arranged on the table so that the sum of the distances from the center of the table to the ends of the minute hands is always greater than the sum of the distances from the center of the table to the centers of the watches.
I don't know if this is correct:
Understanding it geometrically, we can draw a straight lines from the center of table to the center of every watches labelling them $L_1$ to $L_N$. For each line, draw a circle using the line itself as the radius and the center of table as the center. The points where the minute hand distance is greater than center to center distance is where the minute hand touches the circumference of the circle; there are two such points for every circle. The arc formed by the points, and their angles must be depended on the length of the minute hand.
If we look at this using trig functions, I think we can get a function for the length difference for each circle like below, assuming for this watch the distance to center is 1.5 and minute hand length is 1:
$$y=(\sin^2(x))+(1.5-(\cos^2(x))^{0.5}-1.5$$
https://www.wolframalpha.com/input/?i=y%3D((sin(x))%5E2%2B(1.5-cos(x))%5E2)%5E0.5-1.5
To solve (1), the sum of all these functions must be greater than 0 for one particular time (one angle x for every equation). Is this correct?
How to solve (2)? Do we need to use Fourier Series? I played around with the equation, if I add another equation but shift angle by $\pi$, I get a function that does not have negative values, or integral value over the cycle is above 0, which means minute hand distance always greater. But this seem to contradict with the geometric way, because the angle between the two points on the circle's circumference will not always be $\pi$ due to different lengths of the minute hands:
Sorry if this is too long. My math is rusty and at high school / college first year level. This might be a dumb question: Fourier transform breaks down a function into sines and cosines - Given the basis function is
$$(L \sin^2(x)+(R-L \cos^2(x)))^{0.5}-R,$$ where L = minute hand, R = center's distance
and we can only vary the phase angle, L and R, could we calculate a the series required to sum to a function that's non-negative, and also calculate the R, L and phase angle to get this series??
Can the number of watches be odd? Is there a geometric way of solving both (1) and (2).
Thanks!