A weird and spooky clock

Out of nowhere, a clock appeared on my desktop:

(you don't have to be able to view the GIF in order to solve it, I think the "preview" image gives you all the minor clues hidden in it. I also included a text version)

I also found a manual for it:

Hour-hand is the darkest, second-hand the lightest. While all clock hands are exactly 1 long, only the second-hand is independent from others. It likes to tan under high degrees on the floor. Between the hour- & minute-hand is an angel. Anyway, always remember:

A Sane Clock Isn't Interesting

And this clock surely isn't sane :)

Here are all displayed times in standard form (i.e. if the clock face looked like this) in the format hh:mm:ss, but with the start/end positions (which were 03:15:15) excluded:

02:02:33
10:40:24
00:52:07
05:16:51
07:29:57
02:54:05
09:26:28

To check if your solution is most likely correct: The intended plaintext is supposed to tell you the actual time.

The text is

3:15 am

While all clock hands are exactly 1 long

This is implying the unit circle, which has a radius of 1 (with no units)

the second-hand is independent from others. It likes to tan under high degrees on the floor.

This is clever. Floor has two meanings in this sentence. 1) the floor is parallel to the 3 to 9 lines on a clock (or 90° and 270°). We should use the high degrees, or 270. Second, we should floor the answer after we take the tangents.

Between the hour- & minute-hand is an angel

Take the angle between hour and the minute hand

A Sane Clock Isn't Interesting

Use ASCII

To find degrees I used $hourDegrees = hour \times 5 \times 6$ and minutes and seconds use the same for $degrees = value \times 6$

Putting it all together

02:02:33

$60-12 = 48$
$\lfloor\tan(270-198)\rfloor = 3$
$48+3 = 51$
$51$ in ASCII: 3

10:40:24

$300-240 = 60$
$\lfloor\tan(270-144)\rfloor = -2$
$60-2 = 58$
$58$ in ASCII: :

00:52:07

$0 - 312 = -312 = 48$
$\lfloor\tan(270-42)\rfloor = 1$
$48+1 = 49$
$49$ in ASCII: 1

05:16:51

$150 - 96 = 54$
$\lfloor\tan(270-306)\rfloor = -1$
$54-1 = 53$
$53$ in ASCII: 5

07:29:57

$210 - 174 = 36$
$\lfloor\tan(270-342)\rfloor = -4$
$36-4 = 32$
$32$ in ASCII: [space]

02:54:05

$60-324=-264=96$
$\lfloor\tan(270-30)\rfloor = 1$
$96+1 = 97$
$97$ in ASCII: a

09:26:28

$270-156=114$
$\lfloor\tan(270-168)\rfloor = -5$
$114-5 = 109$
$109$ in ASCII: m

• Wait, wait. Doesn't this way of computing the angles assume that the hour hand is always pointing exactly to one of the hours and jumps there as soon as the minute hand crosses the 12? I've never seen an analogue clock that does that. ... HMm, now I look I see that the GIF has exactly that behaviour. That's a little bit evil, and in particular it means that it is not possible to solve the puzzle correctly using the text version. – Gareth McCaughan Oct 31 '16 at 21:52

Here are my thoughts, maybe someone can build full solution on them.

Between the hour- & minute-hand is an angel. means the angle between the hands is important. Angles (in degrees, less than 180 deg) are:
48, 60, 48, 54, 36, 96, 114
A Sane Clock Isn't Interesting means we have to use ASCII table

I don't know what to do with the second-hand yet.

I tried to add the value it points to the angles above and then reading the value of the ASCII table, but nothing worth mentioning came out.

• the "angle - angel" reminds me of Hot Fuzz ^^ – lois6b Oct 31 '16 at 12:13
• I guess, you need to check the tangent value to the second hand.. ("tan under high degrees").. I didn't check myself.. So, chances are it is a dead end... – Sid Oct 31 '16 at 12:15
• Taking the angle between minute and hour hand is tricky. Assuming the hour hand jumps from a to a+1 instantly at the second the hour changes you get the numbers you put, but if you assume it moves 1/60 of tick between a and a+1 every minute we get different numbers. But these numbers aren't as clean(07:29 is not a nice whole numbered angle), so I am not sure why I'm still writing this. – Scott Oct 31 '16 at 17:25
• If you add or subtract from each value an integer less than 6, you can get d:dd am where d represents a digit. "tan under high degrees on the floor" sounds like the variable that determines how much to add to each value. – ETHproductions Oct 31 '16 at 17:36

The text is

3:15 am

Why?

"Between the hour- & minute-hand is an angel."

At each time, take the angle between the hour- & minute-hands,

"Only the second-hand is independent from others. It likes to tan under high degrees on the floor."

subtract the clockwise tangent of the angle between the second-hand and the "floor" (90° angle, or 15 minutes) and floor the result (double-meaning),

"A Sane Clock Isn't Interesting."

and interpret the result as an ASCII char. This results in the string 3:15 am.

• This should be the accepted answer - it's correct; the explanation, while not as thorough as Scott's, includes everything that's needed; and it preceded the other by ~25 minutes. – Guntram Blohm Oct 31 '16 at 18:58
• @GuntramBlohm: Time isn't the only deciding factor for acceptance. It's entirely up to the OP, and "most thorough explanation" is a perfectly reasonable criterion. – Deusovi Nov 1 '16 at 18:09

The words "It likes to tan under high degrees on the floor" suggests to me one of the following:

The tangent of the angle between the second hand and the floor is significant; or the tangent of some other related angle (e.g., between second hand and vertical) is significant; or the tangent of the number of seconds, treated as a number of degrees is significant.

However,

here are the numbers I get, in which I see nothing interesting at all. Each line has: number of seconds; "bearing" in degrees clockwise from noon; tangent of #seconds; tangent and cotangent of "bearing". 33 198 0.6494 0.3249 3.0777 24 144 0.4452 -0.7265 -1.3764 07 42 0.1227 0.9004 1.1106 51 306 1.2349 -1.3764 -0.7265 57 342 1.5399 -0.3249 -3.0777 05 30 0.0875 0.5774 1.7321 28 168 0.5317 -0.2126 -4.7046 I was hoping to see something looking like ASCII codes appearing prominently in one of those columns, but I don't.

• "Tan under high degrees" seems like it only wants us to take a tan of high degrees. But the question still remains: what can be considered as high degree(s)? – jaydm26 Oct 31 '16 at 14:46

I have found that if you arctan the obtuse angle between the minute and hour hands you always get an answer of around 89, though I am not sure of the significance of this...

• The significance is that the arctan of anything large is just a little under one right angle. – Gareth McCaughan Oct 31 '16 at 21:53