A347040 Times on a 12-hour digital clock with 6 digits at which the three continuously moving hands of an analog clock, in the best approximation, enclose the same angles with one another, i.e., have the smallest sum of squares of the deviations from 120 degrees. When interpreting the terms as times of the day in the form hh:mm:ss, padding to the left with zeros is assumed.

2142, 4324, 12647, 14929, 23253, 25435, 33758, 35940, 44404, 50546, 54909, 61051, 65414, 71556, 80020, 82202, 90525, 92707, 101031, 103313, 111636, 113818, 122142, 124324

Offset: 1

Author

Hugo Pfoertner, Aug 13 2021

Keywords

• nonn
• base
• fini
• full
• easy

Comments

An exact hit in which all angles are exactly 120 degrees is impossible. The smallest possible deviation occurs at two points in each display cycle, namely at 02:54:34.5617..., and at 09:05:25.4383... . With rounding to the nearest integer second, this corresponds to the terms a(6)=25435 and a(17)=90525.

The least squares clock solution actually occurs at an exact rational time, namely 5333364000/509173 seconds after 00:00:00, or 285998/509173 seconds after 02:54:34; and the exact least squares sum (in units of squared rotations) is 1/3055038 or (in units of squared clock-second-ticks) 3600/3055038 = 600/509173. - Robert B Fowler, Oct 29 2021

Links

• Henry Ernest Dudeney, Amusements in Mathematics, London, New York, Nelson, 1917. (Problem #63: The Stop-Watch)

Crossrefs

Cf. A120500, A347039, A348637, A350141.

Extensions

a(8) corrected by Robert B Fowler, Oct 29 2021