Problem 891

[FransM]

Problem 891 (View Problem)

If there is an ambiguous position, like in the example, does that count for 1 or for 2. I'm assuming the former but maybe this should be made more explicit.

Also maybe it should be stated that readings are second accurate. Otherwise the problem is insolvable as lots of rotations would result in a good reading (e.g 3:12:35.234)

[RobertStanforth]

Hi FransM, thank you for your questions.

The problem explicitly asks for the number of moments at which the position is ambiguous, not the number of positions. I hope that clarifies things.

As for accuracy, there are no constraints on times being a whole number of seconds, or rounded to the nearest second. The second hand moves continuously.

[utomaya]

In the example,1:30:00 and 7:30:00, are counted as two moments or one moment?

[RobertStanforth]

Two. A "moment" means a "point in time", and these are different points in time.

[wim59]

“Several” ambiguous moments? Hmmm…

[apopov]

Hello!

When the time is let's say 02:38:45, does the seconds hand have an affect on the hour hand? Because I am looking at an analog clock and the hour hand moves only on a completed minute.

Thank you!

[wim59]

… I am looking at an analog clock and the hour hand moves only on a completed minute.

From the problem statement: “All hands look identical and move continuously.”

[mcshaman]

I don't understand how the rules and examples of the problem makes sense:

Despite the inconvenient design, for most time it is possible to tell the correct time (within a 12-hour cycle) from the clock, just by measuring accurately the angles between the hands. For example, if all three hands coincide, then the time must be 12:00:00.

If all the hands are moving all the time, that suggests that the seconds hand is not always in an exact second position meaning that the hands will align about every 1hr and 5 mins. e.g. the hands will all align again some time between 1:05 and 1:06. What am I missing here?

[RobertStanforth]

If all the hands are moving all the time, that suggests that the seconds hand is not always in an exact second position meaning that the hands will align about every 1hr and 5 mins. e.g. the hands will all align again some time between 1:05 and 1:06. What am I missing here?

The Hour and Minute hands will indeed align between 1:05 and 1:06, more precisely at about 1:05:27. At that instant, though, the Second hand is not aligned with the others.

[mcshaman]

If that is what the problem is alluding to then they shouldn't say all the hands are "constantly moving". They should say the second hand is moving a step every second.

So basically this:



Not this:



Thank you for clarifying

[RobertStanforth]

If that is what the problem is alluding to then they shouldn't say all the hands are "constantly moving". They should say the second hand is moving a step every second.

No, that is not the case. The three hands are all continuously moving as per your second diagram. There are no "ticks". The Hour and Minute hand coincide instantaneously at approximately 1:05:27, at just after the "1" position on the dial, but the Second hand is nowhere near at that instant: it's between the "5" and "6" on the dial. It's true that the Second hand was in the vicinity about 22 seconds earlier (and will be again about 38 seconds later), but the Hour and Minute hands do not quite coincide then.

[mcshaman]

Right. Yeah that makes sense now that I think about it. Thank you for the clarification.

[apopov]

If two "moments" in the form $HH_1:MM_1:SS_1$.XXX and $HH_2:MM_2:SS_2$.YYY are ambiguous by the problem definition, do we account for them in the solutions even though when we round up the "moments" to the seconds, we end up with different angles between the hands? Or we can consider a "moment" to have the form : 02:23:53.3312 ?

[RobertStanforth]

If two "moments" in the form $HH_1:MM_1:SS_1$.XXX and $HH_2:MM_2:SS_2$.YYY are ambiguous by the problem definition, do we account for them in the solutions even though when we round up the "moments" to the seconds, we end up with different angles between the hands? Or we can consider a "moment" to have the form : 02:23:53.3312 ?

A "moment" is simply an instant in time, and can indeed have the form that you describe. It is neither constrained nor rounded to a whole number of seconds. The hands move continuously without ticking.

[apopov]

If two "moments" in the form $HH_1:MM_1:SS_1$.XXX and $HH_2:MM_2:SS_2$.YYY are ambiguous by the problem definition, do we account for them in the solutions even though when we round up the "moments" to the seconds, we end up with different angles between the hands? Or we can consider a "moment" to have the form : 02:23:53.3312 ?

A "moment" is simply an instant in time, and can indeed have the form that you describe. It is neither constrained nor rounded to a whole number of seconds. The hands move continuously without ticking.

Thank you!

[defycult]

I have doubts regarding the correctness of the answer. I have briefly commented on this in the discussion thread. I am probably wrong but it would mean a lot of someone could double check. Thanks

[philiplu]

I have doubts regarding the correctness of the answer. I have briefly commented on this in the discussion thread. I am probably wrong but it would mean a lot of someone could double check. Thanks

I posted my thoughts in the forum, and think that maybe the problem statement should be amended to cover your point.

[defycult]

I posted my thoughts in the forum, and think that maybe the problem statement should be amended to cover your point.

Sure I didn't want to clutter the discussion thread. I am not sure if PE tracks past submissions, but if they do, it could be good to retroactively award the question as solved to people who submitted the correct answer according to my interpretation, which may be different from the answer I gave. This is unlikely to include many people and I do agree the question could be clarified a bit.

[FransM]

Is this about this section in the problem

Note that even if two hands perfectly coincide, we can still see them as two distinct hands in the same position. Thus for example 3:00:00 and 9:00:00 are not ambiguous moments.

I don't recall this being there when I first read the problem (but maybe I forgot).

If not can someone enlighten us? Maybe I'm barking at the wrong tree....

[RobertStanforth]

Yes, that is a recent amendment. In light of the points raised in this thread, the Development Team have discussed and rolled out a clarification to the wording for the case where exactly two hands coincide. Thank you @defycult and @philiplu for flagging this to us.