Puzzle ZDJB 

Workings

Hint
In order for the clocks to show the same time (e.g. 2 o'clock), what must the total time gained by one, and lost by the other, total?

Answers
Midnight, 10 days later, when they will both show 4 o'clock.
Midnight, 30 days later, when they will both show 12 o'clock.
Reasoning #1
In order for the clocks to show the same time, the total time gained by one, and lost by the other, must be 12 hours.

For example, if the first clock were to show 2 o'clock, it would have gained 2 hours. In order for the second clock to also show 2 o'clock, it would have had to have lost 10 hours. This is a total of 12 hours gained and lost.

We know that for every hour that passes, the first clock gains one minute, and the second clock loses 2 minutes, for a total time gained and lost of 1 + 2 = 3 minutes.

The total time gained and lost will equal 12 hours when 12 x 60 ÷ 3 = 240 hours have passed. 240 hours = 10 days.

The first clock will have gained 240 x 1 minutes = 240 minutes = 4 hours.

The second clock will have lost 240 x 2 minutes = 480 minutes = 8 hours.

So, they will both show 4 o'clock, 10 days later.

Reasoning #2
In the first answer, we can see that 10 days later, the clocks both show 4 o'clock.

If we move forward another 10 days, both clocks would show 8 o'clock.

If we move forward another 10 days, both clocks would show 12 o'clock.

This will now be the correct time.

So, they will both show 12 o'clock, 30 days later.