Staring at My Watch

~tetris

The trains have been incredibly bad these past few weeks, a fact that makes me mourn even more the current loss of my bike, especially since the WiFi consistently isn't working meaning that there is very little for me to do except stare at my watch and sigh.

To decrease my alarmingly upsetting levels of boredom and my apathy for bureaucracy and people in general, I began to play with my watch on my phone and ponder:

    At what Time(s) are the Minute hand and Hour hand exactly 180°?

What?

For example, at 6pm/6am, the hour hand is exactly facing down and the minute hand exactly facing up. What other instances of this are there?

I came up with two approaches, one brute force and the other more elegant.

1. The Brute Force Approach

• Method: Test every single Minute against every single Hour and measure the angle between them at each point.

• Implementation: I did this in Emacs Lisp (just because!) so you need to copy this into a buffer somewhere (*scratch*?) and evaluate it.

Code

(with-current-buffer (switch-to-buffer "*Messages*")
  (require 'cl)
  (let* ((required-angle-diff 180)
         (angle-precision 1)
         (minute-precision 0.01)
         ;; calc-once
         (angle-between-hours (/ 360 12))
         (all-mins (reverse (cdr (reverse (number-sequence 0 60 minute-precision)))))
         (all-hours (number-sequence 1 12)))
    (cl-labels ((angle-for-min (mins) (* 360 (/ mins 60)))
                (angle-for-hour (hour mins)
                                (let* ((start-hour-angle (* 360 (/ (float (mod hour 12)) 12)))
                                       (min-angle-suppl (* angle-between-hours (/ (float mins) 60))))
                                  (+ start-hour-angle min-angle-suppl)))
                (angle-in-range (diff-angle)
                                (and (> diff-angle (- required-angle-diff angle-precision))
                                     (< diff-angle (+ required-angle-diff angle-precision))))
                (diff-angle-hourmin (hour mins)
                                    (let* ((angle-min (angle-for-min mins))
                                           (angle-hour (angle-for-hour hour mins))
                                           (diff-angle (abs (- angle-hour angle-min))))
                                      (if (angle-in-range diff-angle)
                                          (cons (abs (- diff-angle required-angle-diff))
                                                diff-angle)))))
      ;; Brute force it
      (dolist (curr-hour all-hours)
        (setq hour-hit-min (cons 100 0))
        (dolist (curr-min all-mins)
          (let ((angle-hit (diff-angle-hourmin curr-hour curr-min)))
            (if angle-hit
                (if (< (car angle-hit) (car hour-hit-min))
                    (setq hour-hit-min
                          (list (car angle-hit) curr-min (cdr angle-hit)))))))
        (message "%02d:%02d:%02d - %.2f°"
                 curr-hour
                 (floor (cadr hour-hit-min))
                 (* 60 (- (cadr hour-hit-min) (floor (cadr hour-hit-min))))
                 (caddr hour-hit-min))))))

Result

01:38:10 - 179.99°
02:43:38 - 180.02°
03:49:05 - 180.00°
04:54:33 - 180.03°
05:59:59 - 179.94°
06:00:00 - 180.00°
07:05:27 - 180.02°
08:10:54 - 180.00°
09:16:21 - 180.02°
10:21:49 - 179.99°
11:27:16 - 180.01°
12:32:43 - 180.02°

Interestingly, Hour 5 and Hour 6 share the same position -- meaning there are only 11 states and 22 times a day when the Hour hand and Minute hand are at parity with each other.

2. The Elegant Approach

• Method: Use Logic and Maths to justify why those times

• Implementation: Rambling under my breath at concerned passengers

Observations:

In 1 hour:
* the Minute hand travels 360°
* the Hour hand travels 30°                                                              (Proof: 360 / 12)
* therefore: the Minute hand moves 12° for every degree of the Hour hand                 (Proof: 360 / 30) 

Observations (in sequential order):
* by the time the Minute hand has traveled 360°, the Hour hand has moved an extra 30°    (Proof: 360 / 12)
* the Minute hand needs to travel another 30° to be at parity with the Hour hand.
* by the time the Minute hand has traveled 30°, the Hour hand has moved an extra 2.5°    (Proof: 30 / 12)
* the Minute hand needs to travel another 2.5° to be at parity with the Hour hand.
* by the time the Minute hand has traveled 2.5°, the Hour hand has moved an extra 0.208° (Proof: 2.5 / 12)
* the Minute hand needs to travel another 0.207° to be at parity with the Hour hand.
* ...and so forth....

In total then, the Minute hand needs to move an extra:

• 30 + 2.5 + 0.208 + .... = roughly 32.7 degrees

Deriving the Formula

The above is a Geometric Series of the form:

• Sum (k=0 → Inf) { a r^k }
• where a = 30, and k = 1/12 in this case

The infinite Sum of such a series (convergent if r < 1):

• Sum = a / (1-r) = 30 / (1 - (1/12)) = 360 / 11
• 360 / 11 = 32.727272… ° or 5.454545… mins
• 5 mins 27.3 seconds

So the Minute hand needs to travel 65mins and 27.3 seconds in order to achieve a constant angle between the Hour hand.

To calculate where the Minute hand will be for a given Hour, we just note that the "zero" state of the Minute hand starts at 6, and with this we can generate a general formula of:

         Minute Hand Angle (h) =  ((h - 6) mod 12) * 360 / 11
         Minute Hand Minutes (h)  = ((h - 6) mod 12) * 60 / 11

where h is a given hour

Reflections

None. I got home a few minutes ago. I will watch TV and fall asleep.