A Second Hand Question

" It is a very simple question", said Grandpa

" Look at that wall clock and tell me exactly how many times the Second Hand passes directly over the Minute and the Hour Hands in a 24 hour period. Give me the total times the Second Hand is directly on top of the Minute Hand plus the total times it is directly on top of the Hour Hand"

Simple? I was not convinced.

• riddle tag instead of mathematics??? @DEEM Very suspicious... Simple? I was not convinced – Omega Krypton Feb 3 at 14:45
• There was no Time Measurement Tag :) – DEEM Feb 3 at 14:47
• how about the time tag? – Omega Krypton Feb 3 at 14:51
• I believe that to solve for a given clock we need to know which hands move in a continuous fashion, and for those which don't what their increments are. – Jonathan Allan Feb 3 at 15:50
• Is the clock plugged in? – Acccumulation May 9 at 17:57

Grandpa seems to be laying a lot of stress on the word directly. Let's try this:

2852 $$= (1440 − 24) + (1440 − 2 - 2)$$, where 1440 is the total number of sweeps of the second hand, and 24 and 2 are due to the minute and hour hands moving away from the second hand. The final 2 excludes the hour hand count at exactly 12 o'clock when all three hands are aligned, with the intervening minute hand preventing the second hand from being directly on top of the hour hand.

• Are you sure about that number $22$? I don't think it happens that often on a clock with three hands. Actually, I am assuming the hands are very narrow. In real life with fairly wide hands you may well be right. – Jaap Scherphuis May 8 at 9:51
• I think it's right, isn't it? Just after 5 past 1, just after 10 past 2, just after quarter past 3, and so on for 24 hours. Or have I misunderstood your point? – MichaelMaggs May 8 at 9:56
• Assuming the hands sweep in a continuous motion, at the exact time when the hour and minute hands coincide, the second hand is somewhere else (except for noon/midnight). Of course, if you then go back or forward until the second hand coincides exactly with the hour hand, the minute hand will have moved slightly, but probably not enough to invalidate your argument. – Jaap Scherphuis May 8 at 9:59
• Ah, OK I see what you mean. You may well be right! I've edited my answer. – MichaelMaggs May 8 at 10:00

I'll go with

$$(1440 - 24) + (1440 - 2) = 2854$$

The numbers are the daily full circle counts for the hands, and the hour and minute hands' signs are like that, because they are slowly running away from the second hand.

@Bass has given the sensible answer. But Grampa is probably trolling us. So I'll go with

twenty-six

because I figure that it's

not a mechanical wall clock, it's a digital display made to look like one. From the picture it looks like the hour and minute hands do show part hours/minutes, but the second hand is pointing directly at 9. Probably the display increments once a second, but in steps.

If that's correct, it follows that

the hands do not pass over each other, like on a mechanical clock, unless they are on a whole step. The minute hand will only be on a whole step when the second hand is at 12, and the hour hand will only be on a whole step when both minute and second hands are at 12. Thus, the second hand only passes over the minute hand on the hour (twenty-four times a day), and the hour hand at noon and midnight (twice a day).

• Arguably, it doesn't "pass" over unless it's continuous. – Acccumulation May 9 at 17:58