The clock setter

[BMAD]

Stefan was on his way out the door to visit an old friend across the village when he realized that his grandfather clock had stopped and no longer displayed the correct time.  This was the only clock in his home and the man owned no watches or other time-telling devices.  Without disappointment Stefan left his home and walked roughly three miles to his friend's house.  He glanced at the friend's wall clock as he entered the house and after visiting for a few hours set off back home along the same route.  He walked at the same pace home and had no idea of knowing how long his trip back took him.  Regardless, when Stefan got back home he immediately went to his grandfather clock and set it to the correct time.  How did he know what time it was?

[Pickett]

Spoiler

1. He took note of the time on his clock when he left. We'll call this value T_{S '}
2. He walked x amount of time to his friends house.
3. He took note of the time there. We'll call this value T_{F '}
4. As he leaves he notes the time on his friend's clock. We'll call this value T_{F ''} 
5. He knows what time he arrived and what time he left, so he knows he spent y amount of time at his friend's house: 
   y = (T_{F ''} - T_{F '})
6. He then walks x amount of time back home and looks at his clock. This is T_{S ''}. He knows this value would be: 
   T_{S '' =} T_{S '} + y + 2x
7. So with that information, he can determine x (The amount of time to walk home from his friend's house)  
   x = (((T_{S'' -} T_S') - (T_F'' - T_F')) / 2)
8. He knows the CORRECT current time would be: 
   T_{F ''} + x
9. Therefore, he has all the info he needs to set the correct time:
   T_F'' + (((T_{S'' -} T_S') - (T_F'' - T_F')) / 2)

QUICK EXAMPLE:

• Let's say his clock showed 12:00 when he left (T_{S '} = 12:00)
• His friend's clock showed the actual current time of 17:00 when he arrived. (T_F ' = 17:00)
• His friend's clock showed the actual current time of 20:00 when he left. (T_{F ''} = 20:00) therefore y = 3 hours
• His clock shows 17:00 when he arrives home (T_{S ''} = 17:00)
• He knows he should set his clock to:
  • 20:00 + (((17:00 - 12:00) - 3 hours) / 2)
  • 20:00 + ((5 hours - 3 hours) / 2)
  • 20:00 + (2 hours / 2)
  • 21:00

 

Edited November 30, 2017 by Pickett
Colored the formulas to make it easier to read/follow.

[rocdocmac]

Did Stefan only glance at his friend's wall clock as he entered the house, but not when he left?

[BMAD]

Twice

[Quantum.Mechanic]

Spoiler

"He took note of the time on his clock when he left. We'll call this value T_{S '}"

compared to:

"...when he realized that his grandfather clock had stopped..."

Pickett's answer doesn't seem to acknowledge this (perhaps this was an error in the original statement).