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# 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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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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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:

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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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