# Problem of the week #18 - July 30th, 2012

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#### Jameson

Staff member
This problem was inspired by annoyance of commercials and impatience to wait longer than I absolutely must. If the explanation is unclear please PM me and I'll explain in a different way and make edits to this OP if necessary.

Suppose that you are recording a TV program that is 1 hour long, but there are consistent commercial breaks. The general pattern is 10 minutes of TV followed by 4 minutes of commercials, which you want to fast forward through (skip over them) on your recording device. What is the minimum time, rounded to the nearest minute, into the program you must wait to start watching so that you can skip all the commercials but not catch up to the live program until the very end? The idea here is to end your viewing just as the program is ending but not have to watch commercials.

EDIT: To clarify, when you "skip" the commercials, assume that you the fast forwarding is instant. This isn't how it really works of course but for this problem we'll pretend that when you skip a commercials it's instantaneous.

Remember to read the POTW submission guidelines to find out how to submit your answers!

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#### Jameson

Staff member
Congratulations to the following members for their correct solution this week:

1) Sudharaka

Solution (from Sudharaka):

The program schedule should be as follows. The boxed numbers indicate the time of the TV program whereas the other numbers show the time of the commercials.

$\boxed{10}+4+\boxed{10}+4+\boxed{10}+4+\boxed{10}+4+\boxed{4}=60$

Suppose $$x$$ is the waiting time. If the program should end at the same time at which the recording ends,

$x+44=60$

$\Rightarrow x=16$

So the person will have to wait 16 minutes to start watching.

This is not an accurate model of commercials during a program but the result is very close to what I found through experimenting with different wait times for a 60 minute program. This isn't the most straightforward problem to solve but the result is very interesting and quite logical to me - to skip all the commercials you must wait the length of all the commercials at the beginning, before starting. What you've done is shifted the pattern of program and commercials but not the total time, so instead of $$\displaystyle P+C+P+C...$$ you have $$\displaystyle C_1+C_2...+C_f+P$$