The broken clock

[bonanova]

The minute hand of a clock was lost and had to be replaced. Unfortunately only some extra hour hands were available for use. At how many moments in a day did the repaired clock, with identical hour and minute hands, give an ambiguous time? (Assume AM and PM could be deduced by other means.)

Explanation: At noon (and midnight) the time was certain, even though the hands coincided.  Also, a few minutes after these times, in fact at most other times, it can be deduced which is the minute hand. But there are certain hand positions where that's not possible. How many ambiguous moments occur in the time interval [Midnight, Next Midnight)?

[Molly Mae]

My initial assumption, without diving too deep yet, is that it will happen once per hour.  I'll have to verify that, though, since noon/midnight might actually be that time for the 00/12 hour.  I can say almost certainly it's an even number, since any AM ambiguity will be the same for PM.  That's not much of on observation, though.

[bonanova]

In case it helps, here's a clarification and restatement of the question.

There are moments when it is not possible to identify which hand is the minute hand.
An "ambiguous moment" is when that fact makes the time uncertain.

How many ambiguous moments occur in any 24-hour period?

(Assume we always know the angles of the hands as accurately as needed.)

[Molly Mae]

 

More observations:

We don't care how fast the hands actually move, only their relationship and how many times they move in a 24 hour period.  For each 1 full rotation one hand does, the other does 1/12 of a rotation.  We'll divide the clock into degrees instead of minutes, so for each 360^o of one hand, the other does 30^o. Reduce that down to 13:1, then iterate through all possibilities.  Then we'll look for x|y that has a matching y|x.  

Position in degrees:
13 | 1
26 | 2
39 | 3
and so on.

We can continue to do this until the second column reaches 360 and then we can take the first column mod 360 to actually start comparing against itself.

I'm working on that now, in between my busy schedule.  =P

EDIT: Forgot to add that we actually need to double the final result to get a complete 24 hours.

 

Just ran this up in excel:  Probably too large for a post.  Now I just have to find ones with a duplicate inverse.

13	1
26	2
39	3
52	4
65	5
78	6
91	7
104	8
117	9
130	10
143	11
156	12
169	13
182	14
195	15
208	16
221	17
234	18
247	19
260	20
273	21
286	22
299	23
312	24
325	25
338	26
351	27
4	28
17	29
30	30
43	31
56	32
69	33
82	34
95	35
108	36
121	37
134	38
147	39
160	40
173	41
186	42
199	43
212	44
225	45
238	46
251	47
264	48
277	49
290	50
303	51
316	52
329	53
342	54
355	55
8	56
21	57
34	58
47	59
60	60
73	61
86	62
99	63
112	64
125	65
138	66
151	67
164	68
177	69
190	70
203	71
216	72
229	73
242	74
255	75
268	76
281	77
294	78
307	79
320	80
333	81
346	82
359	83
12	84
25	85
38	86
51	87
64	88
77	89
90	90
103	91
116	92
129	93
142	94
155	95
168	96
181	97
194	98
207	99
220	100
233	101
246	102
259	103
272	104
285	105
298	106
311	107
324	108
337	109
350	110
3	111
16	112
29	113
42	114
55	115
68	116
81	117
94	118
107	119
120	120
133	121
146	122
159	123
172	124
185	125
198	126
211	127
224	128
237	129
250	130
263	131
276	132
289	133
302	134
315	135
328	136
341	137
354	138
7	139
20	140
33	141
46	142
59	143
72	144
85	145
98	146
111	147
124	148
137	149
150	150
163	151
176	152
189	153
202	154
215	155
228	156
241	157
254	158
267	159
280	160
293	161
306	162
319	163
332	164
345	165
358	166
11	167
24	168
37	169
50	170
63	171
76	172
89	173
102	174
115	175
128	176
141	177
154	178
167	179
180	180
193	181
206	182
219	183
232	184
245	185
258	186
271	187
284	188
297	189
310	190
323	191
336	192
349	193
2	194
15	195
28	196
41	197
54	198
67	199
80	200
93	201
106	202
119	203
132	204
145	205
158	206
171	207
184	208
197	209
210	210
223	211
236	212
249	213
262	214
275	215
288	216
301	217
314	218
327	219
340	220
353	221
6	222
19	223
32	224
45	225
58	226
71	227
84	228
97	229
110	230
123	231
136	232
149	233
162	234
175	235
188	236
201	237
214	238
227	239
240	240
253	241
266	242
279	243
292	244
305	245
318	246
331	247
344	248
357	249
10	250
23	251
36	252
49	253
62	254
75	255
88	256
101	257
114	258
127	259
140	260
153	261
166	262
179	263
192	264
205	265
218	266
231	267
244	268
257	269
270	270
283	271
296	272
309	273
322	274
335	275
348	276
1	277
14	278
27	279
40	280
53	281
66	282
79	283
92	284
105	285
118	286
131	287
144	288
157	289
170	290
183	291
196	292
209	293
222	294
235	295
248	296
261	297
274	298
287	299
300	300
313	301
326	302
339	303
352	304
5	305
18	306
31	307
44	308
57	309
70	310
83	311
96	312
109	313
122	314
135	315
148	316
161	317
174	318
187	319
200	320
213	321
226	322
239	323
252	324
265	325
278	326
291	327
304	328
317	329
330	330
343	331
356	332
9	333
22	334
35	335
48	336
61	337
74	338
87	339
100	340
113	341
126	342
139	343
152	344
165	345
178	346
191	347
204	348
217	349
230	350
243	351
256	352
269	353
282	354
295	355
308	356
321	357
334	358
347	359
0	0
13	1
26	2
39	3
52	4
65	5
78	6
91	7
104	8
117	9
130	10
143	11
156	12
169	13
182	14
195	15
208	16
221	17
234	18
247	19
260	20
273	21
286	22
299	23
312	24
325	25
338	26
351	27
4	28
17	29
30	30
43	31
56	32
69	33
82	34
95	35
108	36
121	37
134	38
147	39
160	40
173	41
186	42
199	43
212	44
225	45
238	46
251	47
264	48
277	49
290	50
303	51
316	52
329	53
342	54
355	55
8	56
21	57
34	58
47	59
60	60
73	61
86	62
99	63
112	64
125	65
138	66
151	67
164	68
177	69
190	70
203	71
216	72
229	73
242	74
255	75
268	76
281	77
294	78
307	79
320	80
333	81
346	82
359	83
12	84
25	85
38	86
51	87
64	88
77	89
90	90
103	91
116	92
129	93
142	94
155	95
168	96
181	97
194	98
207	99
220	100
233	101
246	102
259	103
272	104
285	105
298	106
311	107
324	108
337	109
350	110
3	111
16	112
29	113
42	114
55	115
68	116
81	117
94	118
107	119
120	120
133	121
146	122
159	123
172	124
185	125
198	126
211	127
224	128
237	129
250	130
263	131
276	132
289	133
302	134
315	135
328	136
341	137
354	138
7	139
20	140
33	141
46	142
59	143
72	144
85	145
98	146
111	147
124	148
137	149
150	150
163	151
176	152
189	153
202	154
215	155
228	156
241	157
254	158
267	159
280	160
293	161
306	162
319	163
332	164
345	165
358	166
11	167
24	168
37	169
50	170
63	171
76	172
89	173
102	174
115	175
128	176
141	177
154	178
167	179
180	180
193	181
206	182
219	183
232	184
245	185
258	186
271	187
284	188
297	189
310	190
323	191
336	192
349	193
2	194
15	195
28	196
41	197
54	198
67	199
80	200
93	201
106	202
119	203
132	204
145	205
158	206
171	207
184	208
197	209
210	210
223	211
236	212
249	213
262	214
275	215
288	216
301	217
314	218
327	219
340	220
353	221
6	222
19	223
32	224
45	225
58	226
71	227
84	228
97	229
110	230
123	231
136	232
149	233
162	234
175	235
188	236
201	237
214	238
227	239
240	240
253	241
266	242
279	243
292	244
305	245
318	246
331	247
344	248
357	249
10	250
23	251
36	252
49	253
62	254
75	255
88	256
101	257
114	258
127	259
140	260
153	261
166	262
179	263
192	264
205	265
218	266
231	267
244	268
257	269
270	270
283	271
296	272
309	273
322	274
335	275
348	276
1	277
14	278
27	279
40	280
53	281
66	282
79	283
92	284
105	285
118	286
131	287
144	288
157	289
170	290
183	291
196	292
209	293
222	294
235	295
248	296
261	297
274	298
287	299
300	300
313	301
326	302
339	303
352	304
5	305
18	306
31	307
44	308
57	309
70	310
83	311
96	312
109	313
122	314
135	315
148	316
161	317
174	318
187	319
200	320
213	321
226	322
239	323
252	324
265	325
278	326
291	327
304	328
317	329
330	330
343	331
356	332
9	333
22	334
35	335
48	336
61	337
74	338
87	339
100	340
113	341
126	342
139	343
152	344
165	345
178	346
191	347
204	348
217	349
230	350
243	351
256	352
269	353
282	354
295	355
308	356
321	357
334	358
347	359
0	0

Edited January 31, 2018 by Molly Mae

[Molly Mae]

 

So 286 times the hour and minute hand are interchangeable to make a realistic clock reading.  22 of those times aren't ambiguous, though, since the hands are both on the same position.  So my answer is 264 (which is a whole lot higher than I expected).

Edited January 31, 2018 by Molly Mae

[bonanova]

Having second thoughts about your answer, or difficulty seeing it.

Spoiler

Do the numbers represent degrees for the two hands, and you counted the coincidences? They seem to be multiples of 30, which are the hours. But 12 is the only hour the hands coincide. What am I missing?

 

[Molly Mae]

2 hours ago, bonanova said:

Having second thoughts about your answer, or difficulty seeing it.

  Hide contents

Do the numbers represent degrees for the two hands, and you counted the coincidences? They seem to be multiples of 30, which are the hours. But 12 is the only hour the hands coincide. What am I missing?

 

Yeah, I ended up taking it a step further and moving the minute hand by 1 degree at a time instead of the hour hand.  So the matching columns were a looooot longer.  I couldn't include it in the post because it was too long.

Edited February 1, 2018 by Molly Mae

[Molly Mae]

I should also reiterate that I wasn't looking for x|y where x=y.  Those are the 22 results that I removed, since they weren't ambiguous.  I looked for x|y where there was a complimentary y|x

[bonanova]

Sounds good. FTW!

Spoiler

What might you be able to conclude if you added a distinguishable third hand that moved 12 times as fast as the minute hand? i.e. so that the third hand and the minute hand moved like minute and hour hands?

 

[Molly Mae]

10 hours ago, bonanova said:

Sounds good. FTW!

  Hide contents

What might you be able to conclude if you added a distinguishable third hand that moved 12 times as fast as the minute hand? i.e. so that the third hand and the minute hand moved like minute and hour hands?

 

 

In that situation, the second:minute hand is the same as the minute:hour hand.  There are many more cases to evaluate where the hands could be ambiguous, and you can't just compare them to the already ambiguous cases because it will create new ones (I'm fairly confident on this point).  Knowing how long it took me to do just two hands, I don't think I'd like to run it against three. =P  I wouldn't be surprised to find that there's an easier way to get to the answer, but I went with the way that made sense in my head.  If I ever did do it again, I'd be certain to use a scheme that doesn't involve 360^o and go with something that uses whole numbers while being easier to calculate 1/12 of whatever unit I use.

Of course, now that I've typed this up and see that you used the word "distinguishable" instead of "indistinguishable" I'll have to reassess my whole last paragraph.  Knowing how many seconds has passed wouldn't help at all in that moment, though. would actually help, I believe.  It breaks the minute hand down into yet finer movements to be tracked.  

Now I'm starting to think about the "paradox" of moving across a room (where you should never reach the end since you must cross half the room first, then half the remainder).  There probably isn't actually a parallel between the two, but I can see how you could break the movements down smaller and smaller forever, but I don't think you'll ever come up with more ambiguous moments (you won't, actually).  But yeah, next time I'd like to divide the clock into 24000 tocks and divide those into yet smaller ticks so that one tock is 12x ticks and come up with some reasonable x that doesn't make the working numbers too small or large (just for convenience).

Now I feel as though I've rambled.

Edited February 2, 2018 by Molly Mae

[bonanova]

Sorry, I should have pointed out this was a thought experiment and all three hands are distinguishable.

Spoiler

Call the hands H(our), M(inute) and F(ast).

Because on our clock the pairs of hands behave similarly, it gives us 2 time readings: (H, M) and (M, F). It takes all three hands for this except when H=F. Then the two times are given by (H, M) and (M, H).

Aha! Hands H and M give ambiguous times when H=F.

In 24 hours, H rotates 2 times, and F rotates 12x24 = 288 times, giving 286 H=F moments. 286-22=264.