## Friday, June 30, 2017

### The Whimsical Clock Maker (Solution)

Thanks for sending me your solutions! Here is how I went about it.

1.

A whimsical clock maker fashioned both of the hands on his clock to be the same length. Here are some pictures of the clock. What time is it in each picture?

a) 3:00 (It can't be 12:15, because at that time the hour hand wouldn't point directly at the 12.)

b) 4:45 (It can't be 9:23 or what have you, because at that time the hour hand wouldn't point directly at the 9.)

c) 6:30 (It can't be 6:33 or what have you, because at that time the hour hand would be past the 6.)

d) The graphics just aren't very clear in this one—sorry about that. You're right if you said it was ambiguous, or alternatively if you said it was a time right near 11:20, or a time right near 3:56. Sometime I'll try to go back and add hash marks to the clock scale, so that it's more clear exactly where the hands point. The reason I didn't include the hash marks this time around is just that I was pressed for time and reused my old clock graphics code from this post.

UPDATE 7/2/2017: Here is the image of (d) with hash marks. With this detail, we can see that it's 11:20 (not 3:56 and a bit).

e) 6:51 (It can't be 10:34 or what have you, because at that time the minute hand would not be so close to the 10.)

2.

Is the clock maker's clock display ever ambiguous? If so, at what time(s)?

It is indeed possible for the display to be ambiguous. The following pictures show the 66 ambiguous displays. Scroll down below the pictures and I'll briefly explain the method.

UPDATE 7/1/2017: At the bottom of this post I've also added a table of the 66 cases.

The times are shown in a format HH:MM:SS[f], where integer f is the numerator of the fractional part of a second—the denominator is 143 in every case. For example, 00:05:02[14] means 0 hours, 5 minutes, and 2 14/143 seconds.

***

Solution method: Let t be the number of hours past midnight. Then the hour hand and minute hand have angles H(t) and M(t) given by

H(t) = (π/6) t
M(t) = 2π(t − <t>)

where <t> denotes the greatest integer less than or equal to t. (I'm setting things up so that we don't ever deal with angles greater than 2π.)

To find ambiguous clock faces, look for two (distinct) times x and y, such that the hour/minute angle at x equals the minute/hour angle at y:

H(x) = M(y)
M(x) = H(y).

We might as well demand that x > y, otherwise we'd only turn up a raft of redundant solutions (a, b) and (b, a) that actually describe one and the same clock face.

I found it helpful conceptually to express x and y as amounts of time "past the hour," where the hour could be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or 11. Thus I wrote

xm + Δx
yn + Δy

with m, n in {0, 1, 2, … 11}. The simultaneous equations H(x) = M(y), M(x) = H(y) then become, after some simplification,

12Δx = n + Δy
12Δy = m + Δx.

These equations are easily solved to find

Δx = (12n + m)/143
Δy = (12m + n)/143.

The solutions with x > y are those with m > n (to see this, use the solutions for Δx and Δy to show that xy = (12/13)(mn)). So the solutions can be listed as

(m, n) = (1, 0), (2, 0), (2, 1), (3, 0), (3, 1), (3, 2), … (11, 10).

We can count the solutions as 0 + 1 + 2 + … + 11 = (11 × 12)/2 = 66. These solutions give the 66 ambiguous clock faces shown above.

***

00:05:02[14] = 01:00:25[25]
00:10:04[28] = 02:00:50[50]
00:15:06[42] = 03:01:15[75]
00:20:08[56] = 04:01:40[100]
00:25:10[70] = 05:02:05[125]
00:30:12[84] = 06:02:31[7]
00:35:14[98] = 07:02:56[32]
00:40:16[112] = 08:03:21[57]
00:45:18[126] = 09:03:46[82]
00:50:20[140] = 10:04:11[107]
00:55:23[11] = 11:04:36[132]
01:10:29[53] = 02:05:52[64]
01:15:31[67] = 03:06:17[89]
01:20:33[81] = 04:06:42[114]
01:25:35[95] = 05:07:07[139]
01:30:37[109] = 06:07:33[21]
01:35:39[123] = 07:07:58[46]
01:40:41[137] = 08:08:23[71]
01:45:44[8] = 09:08:48[96]
01:50:46[22] = 10:09:13[121]
01:55:48[36] = 11:09:39[3]
02:15:56[92] = 03:11:19[103]
02:20:58[106] = 04:11:44[128]
02:26:00[120] = 05:12:10[10]
02:31:02[134] = 06:12:35[35]
02:36:05[5] = 07:13:00[60]
02:41:07[19] = 08:13:25[85]
02:46:09[33] = 09:13:50[110]
02:51:11[47] = 10:14:15[135]
02:56:13[61] = 11:14:41[17]
03:21:23[131] = 04:16:46[142]
03:26:26[2] = 05:17:12[24]
03:31:28[16] = 06:17:37[49]
03:36:30[30] = 07:18:02[74]
03:41:32[44] = 08:18:27[99]
03:46:34[58] = 09:18:52[124]
03:51:36[72] = 10:19:18[6]
03:56:38[86] = 11:19:43[31]
04:26:51[27] = 05:22:14[38]
04:31:53[41] = 06:22:39[63]
04:36:55[55] = 07:23:04[88]
04:41:57[69] = 08:23:29[113]
04:46:59[83] = 09:23:54[138]
04:52:01[97] = 10:24:20[20]
04:57:03[111] = 11:24:45[45]
05:32:18[66] = 06:27:41[77]
05:37:20[80] = 07:28:06[102]
05:42:22[94] = 08:28:31[127]
05:47:24[108] = 09:28:57[9]
05:52:26[122] = 10:29:22[34]
05:57:28[136] = 11:29:47[59]
06:37:45[105] = 07:33:08[116]
06:42:47[119] = 08:33:33[141]
06:47:49[133] = 09:33:59[23]
06:52:52[4] = 10:34:24[48]
06:57:54[18] = 11:34:49[73]
07:43:13[1] = 08:38:36[12]
07:48:15[15] = 09:39:01[37]
07:53:17[29] = 10:39:26[62]
07:58:19[43] = 11:39:51[87]
08:48:40[40] = 09:44:03[51]
08:53:42[54] = 10:44:28[76]
08:58:44[68] = 11:44:53[101]
09:54:07[79] = 10:49:30[90]
09:59:09[93] = 11:49:55[115]
10:59:34[118] = 11:54:57[129]