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bonanova

The broken clock

Question

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)?

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10 answers to this question

  • 1
 

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 by Molly Mae

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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.

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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.)

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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 360of one hand, the other does 30o. 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 by Molly Mae

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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?

 

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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 by Molly Mae

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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

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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?

 

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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 360and 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 by Molly Mae

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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.

 

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