62.—­THE CLUB CLOCK.

The positions of the hands shown in the illustration could only indicate that the clock stopped at 44 min. 51+1143/1427 sec. after eleven o’clock.  The second hand would next be “exactly midway between the other two hands” at 45 min. 52+496/1427 sec. after eleven o’clock.  If we had been dealing with the points on the circle to which the three hands are directed, the answer would be 45 min. 22+106/1427 sec. after eleven; but the question applied to the hands, and the second hand would not be between the others at that time, but outside them.

63.—­THE STOP-WATCH.

The time indicated on the watch was 5+5/11 min. past 9, when the second hand would be at 27+3/11 sec.  The next time the hands would be similar distances apart would be 54+6/11 min. past 2, when the second hand would be at 32+8/11 sec.  But you need only hold the watch (or our previous illustration of it) in front of a mirror, when you will see the second time reflected in it!  Of course, when reflected, you will read XI as I, X as II, and so on.

64.—­THE THREE CLOCKS.

As a mere arithmetical problem this question presents no difficulty.  In order that the hands shall all point to twelve o’clock at the same time, it is necessary that B shall gain at least twelve hours and that C shall lose twelve hours.  As B gains a minute in a day of twenty-four hours, and C loses a minute in precisely the same time, it is evident that one will have gained 720 minutes (just twelve hours) in 720 days, and the other will have lost 720 minutes in 720 days.  Clock A keeping perfect time, all three clocks must indicate twelve o’clock simultaneously at noon on the 720th day from April 1, 1898.  What day of the month will that be?

I published this little puzzle in 1898 to see how many people were aware of the fact that 1900 would not be a leap year.  It was surprising how many were then ignorant on the point.  Every year that can be divided by four without a remainder is bissextile or leap year, with the exception that one leap year is cut off in the century. 1800 was not a leap year, nor was 1900.  On the other hand, however, to make the calendar more nearly agree with the sun’s course, every fourth hundred year is still considered bissextile.  Consequently, 2000, 2400, 2800, 3200, etc., will all be leap years.  May my readers live to see them.  We therefore find that 720 days from noon of April 1, 1898, brings us to noon of March 22, 1900.

65.—­THE RAILWAY STATION CLOCK.

The time must have been 43+7/11 min. past two o’clock.

66.—­THE VILLAGE SIMPLETON.

The day of the week on which the conversation took place was Sunday.  For when the day after to-morrow (Tuesday) is “yesterday,” “to-day” will be Wednesday; and when the day before yesterday (Friday) was “to-morrow,” “to-day” was Thursday.  There are two days between Thursday and Sunday, and between Sunday and Wednesday.