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

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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

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5:42.3776223776224

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9:23.9160839160839

4:46.993006993007

9:28.951048951049

5:47.4125874125874

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9:39.020979020979

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2:51.1888111888112

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5:52.4475524475525

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6:52.8671328671329

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10:49.5104895104895

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11:54.965034965035

11:4.61538461538462

0:55.3846153846154

11:9.65034965034965

1:55.8041958041958

11:14.6853146853147

2:56.2237762237762

11:19.7202797202797

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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

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11:49.9300699300699

9:59.1608391608392

11:54.965034965035

10:59.5804195804196