1.The full cycle is (12/11 + 12/13 +2) = 574/143 hours elapsed time.

The hour hand advances (12/11+12/13) = 288/143 hours

After 143 such cycles, the hour hand advances 288 hours, which is divisible by 12. (288/12 = 24).

Elapsed time after 143 cycles is 574 hours. And, because the last step of the cycle the hour hand stayed put for exaclty one hour, there was a meeting at the same spot one hour prior. Thus after **573 hours** from the start the hour and minute hand meet at the 12-hour mark.

2. The hour and minute hands also meet at exact hour mark after 65 full cycles plus 12/11 of an hour. (12/11 + 12/13 + 2)*65 + 12/11 = 132.

It so happens that 132 is also divisible by 12. 132/12 = 11

Thus the first meeting at 12 hour mark will take place exactly after (12/11 + 12/13 + 2)*65 + 12/11 = **262 hours. **That is a lot sooner than 573 hours calculated in step 1.

All that is relatively easy to figure out using regular fractions and modular arithmetic. But it does not hurt to verify using the angles in degrees in decimals as calculated by TSLF.

The first step of the cycle the hour hand travels (12/11)*360/12 = 32.72(72).... degrees.

On the third cycle step, the hour hand travels (12/13)*360/12 = 27.69231... degrees.

Making the full cycle: 32.7272(72) + 27.692308 = 60.41958...

After 65 full cycles plus the first step of the next cycle the hour hand travels 60.41958.. * 65 + 32.7272(72) = 3960 (exactly). Which also happens to be divisible by 360. 3960 / 360 = 11.

**I guess this is the closest when both are almost at 12' because (10.9090909090909 rev = 60.41958041958040deg X 65 / 360) : acceptable as 11.

Nice find thanks. I take the more exact (24.0000000000000 rev = 60.41958041958040deg X 143/360 deg) out of my excel tabulation results

And so all 12-hour mark meetings' elapsed times are as following:

262 + 574*n

263 + 574*n

573 + 574*n

574 + 574*n

The first meeting is after **262 hours**.