All entries for Thursday 02 July 2009

July 02, 2009

Covered Clocks

The Warwick Arts Centre has these four clocks:

Clocks in the Arts Centre

(Picture taken from Dilip Mutum's blog)

My initial reaction upon seeing them for the first time was a feeling of bemusement. The function of a clock is to tell the time, so what is the point in hiding half of it? Also, why are the four clocks clustered in one place, rather than spread out nicely in the entire Arts Centre? I eventually realised: it is a work of art. And a clever one, in my opinion. One never needs the entire disc of the clock to be visible in order to tell the time; it is enough to know the position of the two hands. And by placing the clocks together in this fashion, the "artist" made sure that most of the time, it would be enough to look at one clock.

"Most of the time" is the key phrase here. A little thought reveals that sometimes, no single clock shows both hands. If one is only concerned with the hour and the minutes, it is also obvious that suffices to look at two clocks to see both hands. This leads to the question: how often does does one have to look at two clocks to know the time?

As a mathematician, I felt it was my duty to solve this. It turns out that the probability that the probability that two clocks are needed to tell the time, is 0.25. The following argument should make it clear:

A clock is divided into four quarters, or "zones". On every clock, two of these are visible, and two are hidden. By "adjacent zones" I mean zones that are next to each other; by "opposite zones" I mean zones that are not.

For any given hour, the hour-hand is in one of the four zones. If the minute-hand is in the same zone, or in one of the two adjacent zones, there will be a clock showing both hands. Only if the minute-hand is in the opposite zone, then no clock will show both hands. So during every hour, there will be a 15 minute time lapse (when the minute-hand is in the "opposite" zone) during which one has to look at two different clocks. It follows that 1/4 of the time, two clocks are needed. Hence the probability of 0.25.

We have made a few implicit assumptions along the way:

  • One can only tell the time by looking at the long ends of the two hands. Seeing a bit of the short end of a the hour-hand on a clock on whcih the hour-hand is hidden by the black area, is cheating.

  • A hand is never in two zones simultaneously, e.g. a hand cannot be in zone A and zone B in the same time. It will always tend a little to one side. Strictly speaking, this isn't totally true; when the time is, for example, exactly 3 o'clock, the hour-hand will be in zone A and B, while the minute hand will be in zone A and D. However, just one second later, the hour-hand is technically in zone A, and the minute-hand in zone B. My point is, the probabilities of the hands being in two zones are so small (1/3600 for the hour-hand and 1/60 for the minute hand) that we can neglect them. Also, if we view time as something continuous, these probabilities do in fact become zero, but I won't go into that, since it would only serve to confuse everyone.

  • The time on the clocks is uniformly random. In other words, we are equally likely to view the clocks at any time of the day. This isn't realistic. The probability that we are looking at these clocks in the Arts Centre at 3:14 in the middle of the night, is close to nil. But heck, we're setting up an abstract model to figure out this probability, and assuming the time is uniformly random is the most sensible thing to do.

  • The hand showing the seconds isn't needed to figure out what the time is. This assumption is reasonable, and it makes things easier. But it isn't necessary. If one includes the second-hand, we get a new situation which can also be analysed. In this case, the probability that one has to look at (i) one clock, is 0.4375, (ii) two clocks, is 0.5625. Can anyone show this?

But then, this year, something happened. The names of four important cities (New York, Moscow, Beijing and Coventry) were painted underneath the clock, and the time on each clock was then adjusted accordingly.

Clocks in Arts Centre - with names!


I'm sure the individual who made this happen felt that he or she had just shown the signs of a pure genius, and turned 3 redundant clocks into something more practical and business-like. In reality, he or she simply messed up [EDIT: I was wrong about this; see Sarah's comment for more details].  Before the names were painted, telling time was never too problematic, as we've seen. Now, if the hour-hand is not visible in the Coventry clock, one must know the time difference between Coventry and, say, Beijing. Even worse, there are times when the hour hand is visible only on a single clock. Indeed, the visibility of the our-hand goes as follows:

12.00-3.00: Visible only on the Beijing clock
3.00-5.00: Beijing, Coventry
5.00-6.00: New York, Coventry
6.00-9.00: Moscow, New York, Coventry
9.00-11.00: Moscow, New York
11.00-12.00: Beijing, Moscow

(This is during summer time, for the record)

This new layout leads to some new interesting questions:

  1. What is the new probability that one has to look at two clocks in order to see both hour-hand and minute-hand? What if we include second-hand?
  2. If time zones are chosen at random, what is the probability that, sometime during the day, no clocks show the hour-hand? What is the probability that one has to look at two clocks now?
  3. If the each clocks is set at random to any time of the day (so that minutes and seconds don't necessarily match each other on the four clocks), what is the probability that one has to look at two clocks? What if we include the second-hand?
  4. What if, instead of three hands, we have four? To generalise even more, what happens with N hands?

I leave these as an exercise for the reader.

***

Someone must have realised that people were having trouble deriving the actual time from the four clocks. So, as you can see above, each clock was set back to English time. All the clocks say 15:54 in the picture. The funny part is that they have to leave the names of the cities, unless they want to repaint the entire wall white. As a consequence, most people (or at least the ones I've asked) haven't noticed that the clock are all showing the same time.

I guess nowadays most people would just take their mobile phones out to figure out what time it is. I'm old-fashoned in the sense that I still wear a wristwatch.


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