The Time Now is 2:10 p.m. (August 2024 Challenge Question)

Too easy...

Bonus question #1: What time(s) will the minute and hour hands line up (overlap)?

Bonus question #2: Add a second hand. What time(s) will all three hands line up?

Yes, 5° is correct answer for 2:10 because each hour division of the clock is 30° and each 5-minute division of the clock is 30°. So, the calculation is (60°/360°)*30°=5°

The hour and minute hands will line up 12 times a day starting at midnight or noon. The hour hand is 12 times slower than the minute hand so the calculation for the lineup after 1:00 is 5+t/12=t which solves for t=1:27. For after 2:00 the calculation is 10+t/12=t which solves for 2:55. Notice that after 11:00 the equation is 55+t/12=t and solves for noon or midnight

The only solution for the hour, minute, and second hand lineup is noon or midnight. You can calculate any two hands to lineup several times, but not all three except at noon or midnight.

Ratch

The hour and minute hands will line up 12 times a day starting at midnight or noon.

The minute hand goes around 12 times when the hour hand goes around once. The minute hand passes the hour hand 11 times in 12 hours. The second hand goes around 60 times when the minute hand goes around once, passing the minute hand 59 times per hour.

Since 11 and 59 have no common divisor, you are correct, all three hands can only align once every 12 hours, at noon and midnight.

(Written before I looked at any other answers)

At 10 past 2 the minute hand is on the “2” but the hour hand has moved on toward the “3”, where it will reach at 3pm.

The space between 2 & 3 is 5 [time] minutes. Each time-minute on the dial takes 1/60^th of the 360 degrees of the dial, IE 6 degrees, and 5 time-minutes is 30 degrees, which is the space between each number (cross-check 30*12 = 360). 10 minutes is 1/6 of an hour so the hour hand has travelled 1/6 of the distance between 2 and 3. 1/6 of 30 is 5. The minute hand is slap on the 2, the hour hand is 5 degrees further on. The answer is 5 degrees.

Purely dependent on the accuracy of the hand installer at initial construction, last repair or the accuracy of the minute engraver/painter. Anywhere from almost on top of the hour hand through 6deg to somewhere out of this world.

Purely dependent on the accuracy of the hand installer...

How would you like to have this guy's job?

That's what happens with the gifts in cheap Christmas crackers.