Whats the Right Time?

Here is a problem by J.A.H. Hunter.

'That's a fine clock,' Doug remarked, admiring the very modern timepiece on the buffet. 'Electric. It's new,' Steve told him, 'and keeps perfect time.' He raised his wrist. 'I set my watch at noon today by the radio, but it gains just a minute an hour.' Doug nodded. 'I did too,' he said. 'Mine loses two minutes an hour.' Steve's mind was working fast. 'The clock's exactly on a minute division,' he announced, 'so my watch must be showing twice as much after the hour as yours showed after the hour an hour ago.' He's often like that, but both watches clearly needed fixing. So what was the right time?

Please give the answer, if there is one, and show how you got the answer or prove it is impossible to get the answer. By the way this problem is from 1958 so electric clocks would be very modern. Peter Heichelheim

S

P

O

I

L

E

R

At 13:03 (by the electric clock), the fast watch reads 13:04 and the slow watch reads 13:00. But one hour earlier, at 12:03 (by the electric clock), the slow watch read 12:02, so the fast watch is now twice as far past the hour (4) as the slow watch was an hour ago (2).

I did this in Excel.

column C is the electric clock column D is the fast watch column E is the slow watch

row 1 of each column is constant 12:00 (when all three clocks were synchronized)

for subsequent rows

column C =(1/1440)+C1 column D =((61/60)/1440)+D1 column E =((58/60)/1440)+E1

starting at row 61

column G =MINUTE(D61) minutes past hour of fast watch column H =MINUTE(E1) minutes past hour of slow watch an hour ago

C D E F G H 12:00 12:00 12:00 12:01 12:01 12:00 12:02 12:02 12:01 12:03 12:03 12:02 12:04 12:04 12:03 12:05 12:05 12:04 12:06 12:06 12:05 12:07 12:07 12:06 12:08 12:08 12:07 12:09 12:09 12:08 12:10 12:10 12:09 12:11 12:11 12:10 12:12 12:12 12:11 12:13 12:13 12:12 12:14 12:14 12:13 12:15 12:15 12:14 12:16 12:16 12:15 12:17 12:17 12:16 12:18 12:18 12:17 12:19 12:19 12:18 12:20 12:20 12:19 12:21 12:21 12:20 12:22 12:22 12:21 12:23 12:23 12:22 12:24 12:24 12:23 12:25 12:25 12:24 12:26 12:26 12:25 12:27 12:27 12:26 12:28 12:28 12:27 12:29 12:29 12:28 12:30 12:30 12:29 12:31 12:31 12:29 12:32 12:32 12:30 12:33 12:33 12:31 12:34 12:34 12:32 12:35 12:35 12:33 12:36 12:36 12:34 12:37 12:37 12:35 12:38 12:38 12:36 12:39 12:39 12:37 12:40 12:40 12:38 12:41 12:41 12:39 12:42 12:42 12:40 12:43 12:43 12:41 12:44 12:44 12:42 12:45 12:45 12:43 12:46 12:46 12:44 12:47 12:47 12:45 12:48 12:48 12:46 12:49 12:49 12:47 12:50 12:50 12:48 12:51 12:51 12:49 12:52 12:52 12:50 12:53 12:53 12:51 12:54 12:54 12:52 12:55 12:55 12:53 12:56 12:56 12:54 12:57 12:57 12:55 12:58 12:58 12:56 12:59 12:59 12:57 13:00 13:01 12:58 1 0 13:01 13:02 12:58 2 0 13:02 13:03 12:59 3 1 13:03 13:04 13:00 4 2

One last thing, electric clocks don't keep 'perfect' time. A clock whose time base is the 60 Hz frequency of the AC power line will be a minute or two fast or slow. But this error is not cumulative since the power company maintains a long term average of exactly 60 Hz. The error in an oscillator, be it a balance wheel, tuning fork (remember Bulova watches?), or crystal, _is_ cumulative. Thus, watches must be constantly re-synchronized whereas electric clocks do not (if you can tolerate the minor error).

You seem to have assummed that the fast watch jumps a minute immediately after an hour has passed and the slow watch loses a minute immediately after a half hour has passed. This leads to other possible solutions. e.g. 14:08 by the electric clock, 14:10 for the fast watch, and 13:05 for the slow watch an hour ago; 15:13 by the electric clock etc. None of this gives J.A.H. Hunter's answer which I dont understand. So can anyone find the real answer or prove no answer is possible. By the way for the problem assume electric clocks are perfect.

Actually, that's how Excell rounds fractions to the minute, which is reasonable since the problem said nothing about seconds, only minutes.

Could you post J.A.H. Hunter's answer?

That was just a comment about how electric clocks were very modern in 1958. I was implying that being modern, they may not have been well understood, hence the statement 'and keeps perfect time.' Some people get all bent out of shape if you ignore their sensibilities about what constitutes 'reality' in puzzles. I just wanted to make it clear that a 'perfect' clock only exists in 'puzzle physics' and not reality.

Spoiler

. . . . . . . . . . . . . . . . . . . . . . . .

I am assuming that the relationship between Steve and Doug's watches is exact, but that they (unlike the electric clock) are not necessarily showing an exact number of minutes.

If the time past noon in minutes is 60x, then the time on Steve's watch now is 61x, and the time on Doug's watch an hour ago is 58(x-1).

If the former time is twice as far ahead of the hour as the latter, then we have

61x = 116x-116 (mod 60) or 55x = 56 (mod 60),

But this is a dangerously silly equation, since x might not be a whole number. Substituting y=60x, so that y is the whole number of minutes, and multiplying by 12 gives:

11x = 672 (mod 720)

It is easy to check that this is solved today (that is x<720) only when x = 192, so it is now 3:12 PM.

Checking with the original problem:

Steve's watch is now 3.2 minutes fast, so it reads 15.2 minutes after the hour.

An hour ago, Doug's watch was 2*2.2 = 4.4 minutes slow, so it read 7.6 minutes after the hour.

Jonathan Dushoff

Congratulations you got J.A.H. Hunter's answer of 3:12 P.M. and showed it is the only answer. The only problem you should have in your proof that 11y = 672 (mod 720), y<720, and y=192.

Peter