Ambiguous clock puzzle?

This problem actually generalizes the when-do-the-hands-overlap question,

as we'll see below. Let us measure angles clockwise from 12:00,

so a hand pointing at hour 2 is at 60⁰.

Then at m minutes after hour h (written h:m), the minute hand is at 6m⁰

and the hour hand is at (h + m/60)30⁰ = (30h + m/2)⁰.

An ambiguity will occur if there is another time H:M

which interchanges hands with h:m. That is,

6M = 30h + m/2 and 30H + M/2 = 6m; i.e.

M = 5h + m/12 and m = 5H + M/12.

Substituting and simplifying,

M = 5h + m/12 = 5h + (1/12)(5H + M/12) = 5h + (5/12)H + M/144; thus

(143/144)M = 5h + (5/12)H and hence

(143/12)M = 60h + 5H.

Similarly, we obtain

(143/12)m = 60H + 5h.

The last two equations are the source of all the ambiguities...except when h=H.

In that case, the hands overlap, and there is no ambiguity.

Otherwise, for each h,H ∈ {0,1,2,...,11} with h<H, there is an ambiguity.

Here's an example. Take h=2 and H=5. Then the equations are

(143/12)M = 145 and (143/12)m = 310, so

M = 1740/143 = 12 24/143 and m = 3720/143 = 26 2/143.

The two ambiguous times are

5:12 24/143 and 2:26 2/143.

When h=H, we get (143/12)m = 65h, so m = 60h/11, as usual.

RE: DELETED the earlier response where I mistakenly computed the coincidental phases of a clock....

YUP I misunderstood the meaning of ambiguous. Here is my attempt to solve what you asking which seems more tricky.

Now let,

A = angle made by hour-hand with an imaginary axis-line that runs across 12

B = angle made by minute-hand with an imaginary axis-line that runs across 12

Only considering initially time-zone between 12:00 - 1:00 for sake of simplicity, we will try to find the ambiguous time zones.

A will only have a range of 0 - 30 degrees in this time zone

B will make an angle of

B = B' + 30x … (1)

where,

B': Runtime angle made between 0 - 30 degrees and 30x is the accumulation of earlier angles covered by the minute hand.

x: Time slot variable running form 1 to 11

Now, Ambiguity will occur when the proportion of revolution covered by A and B are same when they represent hour and minute hands OR when they are reversed in their functioning. i.e hour becomes minute hand and vice-versa. In algebraic terms

A/30 = B/360

=> A = B/12 … (2)

OR (when clock hand changes role)

A/360 = B'/30 {we used B' since now minute hand has become hour hand so it will only cover max 30 degrees in its movement}

Substitute B' from Eq (1), we get,

A/360 = (B - 30x)/30

A/360 = B/30 - x

Substitute A from Eq (2)

B/12*360 = B/30 - x

B/4320 = B/30 - x

x = B/30 - B/4320

x = 143 B/4320

=>

B = 4320 x / 143

B = 30.209x degrees

Since 30.209 is in degrees, we can divide the number by 6⁰ to get the minutes past since 6⁰ movement makes 1 minute, hence

B = 5.035x minutes

Now by putting the values of x from 1 to 11 we can get first 11 ambiguous time-zones. Some are

x = 1; minutes_past = 5.035; 12:05:02

x = 2; minutes_past = 10.07; 12:10:04

x = 3; minutes_past = 15.105; 12:15:06

.

.

.

In a similar fashion we can generalise the formula for A to cover all the designated hour-zones between NOON - MIDNIGHT

So in all 12 hours are to be covered with each giving 11 ambiguities.

Hence total ambiguous time-zones are 12 X 11 = 132

Watch

one

132

http://trikeshed.org/maths/ambiguous_clock.html