OFFSET
1,2
COMMENTS
Consider an analog clock face to be a unit circle, with unit-length clock hands; the endpoints of the hands lie on the unit circle and form the vertices of a Clock Triangle inscribed within the circle.
The area within this Clock Triangle has maximum value 1.2990353071..., which occurs around 02:54:35 and at its mirror image around 09:05:25.
At time T seconds after 00:00:00, the clock hands are at angles
S (seconds hand) = T/60 * 360, (degrees)
M (minutes hand) = T/60/60 * 360,
H (hours hand) = T/60/60/12 * 360.
The clock cycle repeats every 12 hours = 43200 seconds.
The second 6 hours of the cycle is a mirror image of the first 6 hours.
The area within the Clock Triangle at any time is equal to
F(T) = abs(sin(H-M) + sin(M-S) + sin(S-H))/2.
(The derivation of this equation is not overly-complicated.)
The hour and minute hands are exactly 120 degrees apart at times
T = 14400/11*(3k+1) and T = 14400/11*(3k+2) for integer k.
There are 22 such times during every 12-hour cycle.
Empirically examining the relative extrema of F(T) near these 22 times, it is found that the largest F(T) occurs near T = 10475 (02:54:35), and near its mirror image T = 32725 (09:05:25).
Using Newton's iterative method to solve for Tmax in F'(Tmax) = 0,
Tmax = 10474.561690797181984...
F(Tmax) = 1.299035307107332...
Note: an equilateral triangle has area sqrt(3)*3/4 = 1.2990381056...
REFERENCES
H. E. Dudeney, Amusements in Mathematics, Dover, 1958, pages 11 and 154; Problem #63, "The Stop-Watch", notes that at times 02:54:35 and 09:05:25 the clock hands are "nearly equidistant" and that "exact equidistance for the three hands is not possible", but does not point out that these two times are the most nearly equidistant times in the 12-hour clock cycle. Of the 430 puzzles in the book, 4 are represented on the front cover illustration; the stop-watch appears in the very center.
LINKS
Henry Ernest Dudeney, Amusements in Mathematics, London, New York, Nelson, 1917.
Henry Ernest Dudeney, Cover illustration of Amusements in Mathematics with clock, New York, Dover, 1958.
MATHEMATICA
f[x_] := (Abs[Sin[2*Pi*x*(1/43200 - 1/3600)] + Sin[2*Pi*x*(1/3600 - 1/60)] + Sin[2*Pi*x*(1/60 - 1/43200)]])/2; RealDigits[FindMaximum[f[x], {x, 10475}, WorkingPrecision -> 110][[1]], 10, 100][[1]] (* Amiram Eldar, Oct 27 2021 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Robert B Fowler, Oct 26 2021
STATUS
approved