Between-2-00-and-2-15-what-time-is-it-exactly-when-the-hour-minute-and-second-s-hand-of-a-clock-occupy-the-same-angular-position-

Question Number 17449 by ajfour last updated on 06/Jul/17

Between 2 : 00 and 2 : 15, what time

is it exactly when the hour,

minute, and {second}^{'} s hand of a

clock occupy the same angular

position .

Commented by mrW1 last updated on 06/Jul/17

in the interval [2 : 00 - 2 : 12) the hour

hand remains at 2 : 00 \equiv position 10

since the hour hand moves a step

every 12 minutes .

in the interval [2 : 10 - 2 : 11) the minute

hand remains at position 10, since

the {minute}^{'} s hand moves a step

every 60 seconds .

at 2 : 10 : 10 the second hand is at

position 10 .

Commented by ajfour last updated on 06/Jul/17

yes sir, i understood your answer

(with floor function) only a little

later after i commented .

Commented by mrW1 last updated on 07/Jul/17

I think, if the hands move

continiously, they stay all in the same

position only at 12 : 0 : 0 .

Commented by ajfour last updated on 07/Jul/17

complete research, thank you sir .

Answered by mrW1 last updated on 06/Jul/17

the hands on a clock {don}^{'} t move

continiously .

there are totally 60 possible positions

for the three hands on a clock :

p = 0, 1, 2, \dots 59 .

{let}^{'} s say x hours after 2 o^{'} clock .

position of {hour}^{'} s hand is p_{h} = 10 + ⌊ 5 x ⌋

position of {minute}^{'} s hand is p_{m} = ⌊ 60 x ⌋

position of {second}^{'} s hand is p_{s} = ⌊ \frac{3600 x}{60} ⌋

at 2 : 10 : 10

x = 0 h 10 m 10 s = 0.1694 \dots hour

p_{h} = 10 + ⌊ 5 \times 0.1694 ⌋ = 10

p_{m} = ⌊ 60 \times 0.1694] = 10

p_{s} = ⌊ \frac{3600 \times 0.1694}{60} ⌋ = 10

\Rightarrow they are at the same position .

Commented by mrW1 last updated on 06/Jul/17

other times when the three hands of a clock

are at the same position :

12 : 0 : 0

1 : 5 : 5

2 : 10 : 10

3 : 16 : 16

4 : 21 : 21

5 : 27 : 27

6 : 32 : 32

7 : 38 : 38

8 : 43 : 43

9 : 49 : 49

10 : 54 : 54

11 : 59 : 59

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