## Thursday, November 19, 2009

### Long Dinner

You met a few old friends for dinner at the restaurant in between 6 and 7pm. Time flew and you did not notice how hours passed. The only strange thing is that when you got out of a restaurant past 9pm and looked at your watch, you saw that the minute and hour hands have exchanged places. Were you in a restaurant for more than 3 hours, if you got home at 10pm?
(idea is taken from B. Kordemsky book)

Submit your answer on our Family Puzzle Marathon site. Solve three puzzles and get a prize!

Kim said...

There's not enough information to tell.

If you got out between 9 and 10, then no you weren't there for more than three hours. You would have had to have arrived between 6:45 and 6:50 and left around 9:30. Less than three hours.

But the wording of the question would allow for you to have arrived at the restaurant at, for example, 6:55 and left the restaurant at about 11:30, which would be more than 3 hours.

Maria said...

Oooops, I missed an important detail - corrected it now, by adding that you reached home at 10pm.

Thank you, Kim. You are right - less than three hours! A point for Kim. I'll give another puzzle point to anyone who can shake our rusty brains and figure out the exact in-and-out times.

Kim said...

To get the hour hand in the same spot as the corresponding minute hand, we need:

12x = 45+x
11x = 45
x = 4.09

So, it's at 6:49.054

Kim said...

That is: 6:49 +5.4 seconds

Kim said...

Sorry, that's just the "in" time. The "out" time then would be 9:34.

Maria said...

Something does not feels right here.
If when person enters the restaurant it is 6.49pm then minute handle is very close to 50mins. This minute handle becomes hour handle and therefore time should be almost 10pm, but we know that the other handle is in-between 6 (30 min) and 7 (35 min), so it can't be as close to 10pm.
It should be just a bit earlier.
Leaving this as a super-challenge for someone to crack.
New puzzle and a newsletter tomorrow.

Barry said...

Hey Math Mom, my answer is:

Arrival Time: 6:47
Departure Time: 9:33

From the puzzle, we know you arrived between 6 and 7, and you departed between 9 and 10.

Since the minute and hour hands switched, then you can narrow down the arrival time to 6:45 -> 6:50 and the departure time to 9:30 -> 9:35 just by looking at the position of the hour hands at the extreme cases (6:00, 7:00, 9:00, and 10:00).

Then examine each case for the arrival time, starting with 6:45.

At 6:45, the minute hand is on the 9 and the hour hand is somewhere between the 6 and 7. If you divide up the distance between the 6 and 7 into 5 discrete hops, then it is 45/60 == 75% of the way from the 6 to the 7, which rounds down to being 3 hops in, or pointing at the 33 minute mark, so we can say that a 6:45 arrival corresponds to a 9:33 departure.

For all of the possible arrival times, the departures correspond to:

6:45 -> 9:33
6:46 -> 9:33
6:47 -> 9:33
6:48 -> 9:34 (It increases to :34 here because 48/60 puts it in the 4th hop)
6:49 -> 9:34
6:50 -> 9:34

The final step is to look at the arrival time minute hand again. At 6:45, the minute hand is pointing directly at the 9. But at 9:33, we know that the hour hand would have started moving towards the 10. The hour hand will only continue to point directly to the 9 between 9:00 and 9:11, and 9:33 falls outside that range, so we know that 6:45 isn't the right arrival time.

Again, applying this logic to all possible arrival times:

9:45 -> 9:33 -> must fall between 9:00 and 9:11
9:46 -> 9:33 -> must fall between 9:12 and 9:23
9:47 -> 9:33 -> must fall between 9:24 and 9:35
9:48 -> 9:34 -> must fall between 9:36 and 9:47
9:49 -> 9:34 -> must fall between 9:48 and 9:59
9:50 -> 9:34 -> must fall between 10:00 and 10:11

Therefore, the correct arrival time is 9:47.

Maria said...

That is correct, Barry!
You came in at 6:47pm and left the restaurant at 9:33pm. In the last paragraph, you just typed 9 instead of 6. Another great goal for Barry.