# After Five O’clock

This is a fairly extensive clock problem by Geoffrey Mott-Smith from 1954.

“The clock shown in the illustration has just struck five. A number of things are going to happen in this next hour, and I am curious to know the exact times.

1. At what time will the two hands coincide?
2. At what time will the two hands first stand at right angles to each other?
3. At one point the hands will stand at an angle of 30 degrees, the minute hand being before the hour hand. Then the former will pass the latter and presently make an angle of 60 degrees on the other side. How much time will elapse between these two events?”

See After Five O’Clock for solutions.

# Clock Connections Puzzle

This is an imaginative puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings for 2012.

“You draw a line connecting the 5 and 9 on a clock face, and another line connecting the 3 and 8. What is the angle between the two lines?”

See the Clock Connections Puzzle for solutions.

# Grandfather Clock Puzzle

This is another doable puzzle from Sam Loyd.

“BACK OF THE OLDTIME song of “Grandfather’s clock was too tall for the shelf, so it stood for ninety years on the floor,” there was a legend of a pestiferous grand-father and a cantankerous old clock which, from the fitful time when “it was bought on the morn, when the old man was born,” it had made his whole life miserable, owing to an incurable habit which the clock had acquired of getting the hands tangled up whenever they attempted to pass.

These semi-occasional stoppages became of more frequent occurrence as advancing age made the old gentleman more irritable and his feeble hands more incapable of correcting the cranky antics of the balky old timepiece.

Once when the hands came together again and stopped the clock the old man flew into such an ungovernable passion that he fell down in a fit, stone dead, and it was then that

“The clock stopped short,

Never to go again,

When the old man died.”

A photograph of the clock was presented to me, showing the classical figure of a female representing time, and it struck me as remarkable that with the knowledge of the hour and minute hands being together that it should be possible to figure out the exact time at which “the old man died,” from the position of the second hand as shown, without having to see the face of the clock. The idea of being able to figure out the exact time of day from seeing the second hand alone is very odd, although not so difficult a puzzle as one would imagine.”

See the Grandfather Clock Puzzle for a solution.

# Hard Time Conundrum

This problem comes from the “Problems Drive” section of the Eureka magazine published in 1955 by the Archimedeans at Cambridge University, England.  (“The problems drive is a competition conducted annually by the Archimedeans. Competitors work in pairs and are allowed five minutes per question ….”)

“There are ten times as many seconds remaining in the hour as there are minutes remaining in the day. There are half as many minutes remaining in the day as there will be hours remaining in the week at the end of the day.  What time is it on what day?”

One of the hardest parts of the problem is just being able to translate the statements into mathematical terms.  Solvable in 5 minutes?!!!

See the Hard Time Conundrum for a solution.

# A Question of Time

This turns out to be an unambiguous, doable problem from the 19th century puzzle master Sam Loyd.  It is based on an observation about jewelers’ signs of the times.  I thought I would include Loyd’s narrative in its entirety.

“A CURIOUS paragraph has been going the rounds of the press which attempts to explain why the signs of the big watches in front of jewelry stores are always alike. They are painted upon the dial, apparently in a haphazard sort of a way, and yet they invariably indicate a certain number of minutes past eight. It cannot be attributable to chance, for it would tax one’s credulity to believe that such a coincidence could occur all over the civilized world.

There is no accepted rule or agreement established with the jewelers or sign painters, for careful inquiry proves that few of them are aware of the fact or ever noticed that any two are alike, it would be a marvelous case of unconscious imitation if it is looked upon as a mere custom, accidentally following a pattern set by the originator of the device of the sign of a big watch. In London, where they take pride in such things, I saw several big watches, looking as if they had hung in front of the stores for countless centuries, all indicating the same mysterious time, accompanied by the announcement that the firms were established a couple of hundred years ago. I do not doubt for a moment that some such similar sign can be found at Nuremberg, where the watch originated during the Fifteenth Century.

The discussion seems to have brought out a recognition of the fact that from an artistic point of view, symmetry requires that the hands should be evenly balanced, as it were, on both sides of the face of the watch.

If they are raised too much there is a certain “exasperating, declamatory effect,” which is not altogether pleasing.

The time would be incorrect if the hands pointed at 9 and 3, and at other points would be too low, so, as a matter of fact, and from an artistic point of view, the position is well selected and is one of the points which, with the aid of a watch, can be shown to be possible. It is a fact however, that the mere puzzle of telling what time the watch indicates, has been held up to public gaze for all these centuries without being thought of or solved?

Take your watch and set it to the time indicated, with the hands at equal distances from the six hour, which shows it to he a possible position, and then tell what time of the day it is! …”

See A Question of Time for solutions.

# Fallen Clock Puzzle

This is a nice variation on the typical clock problem posed by Cary Mallon and retweeted by Henk Reuling:

“This clock has fallen on the floor, and unfortunately, there is no indication which way ‘up’ the clock should hang.  However, both hands are pointing precisely at the [adjacent] minute marks.  You can now work out what the time is.”

See the Fallen Clock Puzzle for a solution.