All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
How do I know the time? (Posted on 2019-07-20) Difficulty: 3 of 5
The hour and minute hands of a clock are indistinguishable (and there is no second hand). How many moments are there in a 12-hour cycle when it is not possible to tell from this clock what time it is?

No Solution Yet Submitted by Danish Ahmed Khan    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution algebraic analysis and computer solution Comment 1 of 1
We'll assume that the hands' motions are continuous, and that people looking at the clock have perfect visual acuity and ability to assess valid clock positions exactly.

At time k + f, where k is an integral hour and 0<=f<1  is a fraction of an hour, the hour hand is (k+f)/12 around the face and the minute hand is the fraction f around the face.

We seek to know instants when

f = (k'+f')/12
  and
(k+f)/12 = f'

where both k and k' are integers and 0<=f<1 and 0<=f'<1.

f = (k' + (k+f)/12) / 12
  = k'/12 + k/144 + f/144
f*(1-1/144) = k'/12 + k/144

So given k and k' (the integral hours--real and confused with):

f = (k'/12 + k/144)/(1 - 1/144)
f' = (k+f)/12

which are the added fractional hours.

since integers k and k' determine both f and f' there can be only one instant within an hour where the ambiguity occurs for a specific other hour; and if both hours are the same, this refers only to the time at which the hands coincide and there is no real ambiguity.

This program shows and counts the times of true ambiguaity:

 For k = 0 To 11
 For kp = 0 To 11
  If k <> kp Then
   f = (kp / 12 + k / 144) / (1 - 1 / 144)
   fp = (k + f) / 12
   Text1.Text = Text1.Text & k & ":" & f * 60 & crlf
   Text1.Text = Text1.Text & kp & ":" & fp * 60 & crlf & crlf
   ct = ct + 1
  End If
 Next
 Next
 
The output shows 132 times, all valid, as no negative minutes were found. That means for every hour stretch there were times in each of the other 11 hour stretches where a point of ambiguity occurred.

Note that we don't halve the count for symmetry, as, for example, 12:05.03..., confused with 1:00.41..., is a different moment from 1:00.41... confused with 12:05.03....

The list shows hour followed by minute (rather than the f fraction of an hour) and zero hours represents 12:

0:5.03496503496504
1:0.41958041958042

0:10.0699300699301
2:0.839160839160839

0:15.1048951048951
3:1.25874125874126

0:20.1398601398601
4:1.67832167832168

0:25.1748251748252
5:2.0979020979021

0:30.2097902097902
6:2.51748251748252

0:35.2447552447552
7:2.93706293706294

0:40.2797202797203
8:3.35664335664336

0:45.3146853146853
9:3.77622377622378

0:50.3496503496503
10:4.1958041958042

0:55.3846153846154
11:4.61538461538462

1:0.41958041958042
0:5.03496503496504

1:10.4895104895105
2:5.87412587412587

1:15.5244755244755
3:6.29370629370629

1:20.5594405594406
4:6.71328671328671

1:25.5944055944056
5:7.13286713286713

1:30.6293706293706
6:7.55244755244755

1:35.6643356643357
7:7.97202797202797

1:40.6993006993007
8:8.39160839160839

1:45.7342657342657
9:8.81118881118881

1:50.7692307692308
10:9.23076923076923

1:55.8041958041958
11:9.65034965034965

2:0.839160839160839
0:10.0699300699301

2:5.87412587412587
1:10.4895104895105

2:15.9440559440559
3:11.3286713286713

2:20.979020979021
4:11.7482517482517

2:26.013986013986
5:12.1678321678322

2:31.0489510489511
6:12.5874125874126

2:36.0839160839161
7:13.006993006993

2:41.1188811188811
8:13.4265734265734

2:46.1538461538462
9:13.8461538461538

2:51.1888111888112
10:14.2657342657343

2:56.2237762237762
11:14.6853146853147

3:1.25874125874126
0:15.1048951048951

3:6.29370629370629
1:15.5244755244755

3:11.3286713286713
2:15.9440559440559

3:21.3986013986014
4:16.7832167832168

3:26.4335664335664
5:17.2027972027972

3:31.4685314685315
6:17.6223776223776

3:36.5034965034965
7:18.041958041958

3:41.5384615384615
8:18.4615384615385

3:46.5734265734266
9:18.8811188811189

3:51.6083916083916
10:19.3006993006993

3:56.6433566433566
11:19.7202797202797

4:1.67832167832168
0:20.1398601398601

4:6.71328671328671
1:20.5594405594406

4:11.7482517482517
2:20.979020979021

4:16.7832167832168
3:21.3986013986014

4:26.8531468531469
5:22.2377622377622

4:31.8881118881119
6:22.6573426573427

4:36.9230769230769
7:23.0769230769231

4:41.958041958042
8:23.4965034965035

4:46.993006993007
9:23.9160839160839

4:52.027972027972
10:24.3356643356643

4:57.0629370629371
11:24.7552447552448

5:2.0979020979021
0:25.1748251748252

5:7.13286713286713
1:25.5944055944056

5:12.1678321678322
2:26.013986013986

5:17.2027972027972
3:26.4335664335664

5:22.2377622377622
4:26.8531468531469

5:32.3076923076923
6:27.6923076923077

5:37.3426573426573
7:28.1118881118881

5:42.3776223776224
8:28.5314685314685

5:47.4125874125874
9:28.951048951049

5:52.4475524475525
10:29.3706293706294

5:57.4825174825175
11:29.7902097902098

6:2.51748251748252
0:30.2097902097902

6:7.55244755244755
1:30.6293706293706

6:12.5874125874126
2:31.0489510489511

6:17.6223776223776
3:31.4685314685315

6:22.6573426573427
4:31.8881118881119

6:27.6923076923077
5:32.3076923076923

6:37.7622377622378
7:33.1468531468531

6:42.7972027972028
8:33.5664335664336

6:47.8321678321678
9:33.986013986014

6:52.8671328671329
10:34.4055944055944

6:57.9020979020979
11:34.8251748251748

7:2.93706293706294
0:35.2447552447552

7:7.97202797202797
1:35.6643356643357

7:13.006993006993
2:36.0839160839161

7:18.041958041958
3:36.5034965034965

7:23.0769230769231
4:36.9230769230769

7:28.1118881118881
5:37.3426573426573

7:33.1468531468531
6:37.7622377622378

7:43.2167832167832
8:38.6013986013986

7:48.2517482517483
9:39.020979020979

7:53.2867132867133
10:39.4405594405594

7:58.3216783216783
11:39.8601398601399

8:3.35664335664336
0:40.2797202797203

8:8.39160839160839
1:40.6993006993007

8:13.4265734265734
2:41.1188811188811

8:18.4615384615385
3:41.5384615384615

8:23.4965034965035
4:41.958041958042

8:28.5314685314685
5:42.3776223776224

8:33.5664335664336
6:42.7972027972028

8:38.6013986013986
7:43.2167832167832

8:48.6713286713287
9:44.0559440559441

8:53.7062937062937
10:44.4755244755245

8:58.7412587412587
11:44.8951048951049

9:3.77622377622378
0:45.3146853146853

9:8.81118881118881
1:45.7342657342657

9:13.8461538461538
2:46.1538461538462

9:18.8811188811189
3:46.5734265734266

9:23.9160839160839
4:46.993006993007

9:28.951048951049
5:47.4125874125874

9:33.986013986014
6:47.8321678321678

9:39.020979020979
7:48.2517482517483

9:44.0559440559441
8:48.6713286713287

9:54.1258741258741
10:49.5104895104895

9:59.1608391608392
11:49.9300699300699

10:4.1958041958042
0:50.3496503496503

10:9.23076923076923
1:50.7692307692308

10:14.2657342657343
2:51.1888111888112

10:19.3006993006993
3:51.6083916083916

10:24.3356643356643
4:52.027972027972

10:29.3706293706294
5:52.4475524475525

10:34.4055944055944
6:52.8671328671329

10:39.4405594405594
7:53.2867132867133

10:44.4755244755245
8:53.7062937062937

10:49.5104895104895
9:54.1258741258741

10:59.5804195804196
11:54.965034965035

11:4.61538461538462
0:55.3846153846154

11:9.65034965034965
1:55.8041958041958

11:14.6853146853147
2:56.2237762237762

11:19.7202797202797
3:56.6433566433566

11:24.7552447552448
4:57.0629370629371

11:29.7902097902098
5:57.4825174825175

11:34.8251748251748
6:57.9020979020979

11:39.8601398601399
7:58.3216783216783

11:44.8951048951049
8:58.7412587412587

11:49.9300699300699
9:59.1608391608392

11:54.965034965035
10:59.5804195804196
  

  Posted by Charlie on 2019-07-20 12:38:54
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (14)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information