44 pills in 30 days
 

A doctor tells a patient to take 44 pills over a period of 30 days. The patient is free to choose when to take the pills, as long as they take at least one pill a day on each of the 30 days.

Prove that no matter what the patient does, there will be a period of consecutive days where exactly 11 pills are taken.

[spoiler]Another counterexample (you sure you got the question right?): If the patient takes exactly 2 a day until the patient runs out, there will be no period of consecutive days where exactly 11 pills are taken. In fact, if the patient takes a factor of 44 excluding 1 and 11 each day, (2, 4, 22, 44) there will be no period of consecutive days where exactly 11 pills are taken.[/spoiler]
[quote=bsquared;158554]You are both forgetting the above statement in the problem.[/quote]
Is the 'one pill per day' rule a minimum for every day, and the patient must always have at least one available to take that day? (meaning the patient must run out on day 30) In that case, both counterexamples can be ignored, and I don't know for sure how to prove it.
With the way it's worded it seems that 44 pills must be taken within 30 days, at least one per day until you run out, you didn't specify that you must have some through day 30.

[quote=Orgasmic Troll;158550] as long as they take at least one pill a day.
[/quote]

You are both forgetting the above statement in the problem.

[QUOTE=Mini-Geek;158553][spoiler]Another counterexample (you sure you got the question right?): If the patient takes exactly 2 a day until the patient runs out, there will be no period of consecutive days where exactly 11 pills are taken. In fact, if the patient takes a factor of 44 excluding 1 and 11 each day, (2, 4, 22, 44) there will be no period of consecutive days where exactly 11 pills are taken.[/spoiler]

Is the 'one pill per day' rule a minimum for every day, and the patient must always have at least one available to take that day? (meaning the patient must run out on day 30) In that case, both counterexamples can be ignored, and I don't know for sure how to prove it.
With the way it's worded it seems that 44 pills must be taken within 30 days, at least one per day until you run out, you didn't specify that you must have some through day 30.[/QUOTE]

My god, do you really need me to be that pedantic?

[spoiler]
I've got 44 pills laid out in a row on the table, and I've got 30 straws. I'm going to use the straws as dividers between the pills, so that every day I take all the pills up to the next straw. "At least one pill per day" means that I cannot put two straws in the same place, and "take all the pills" means the 30th straw must be after the 44th pill (not actually "between", but you understand).

Now suppose somebody claims to have a solution - an arrangement of the straws with no consecutive days totaling 11 pills.

There cannot be a straw between pill 11 and pill 12.

Let S1 be the number of pills taken on day 1. There cannot be a straw between S1+11 and S1+12 - and this is a different slot because S1>0.

Let S2 be the the sum of the number of pills taken through day2. There cannot be a straw between S2+11 and S2+12. And this is different from the other "blackouts" because S2>S1.

At the end of day 18 there must be at least 12 pills left because you must take on on each the remaining 12 days. Let S18 be the the sum of the number of pills taken through day 18. There cannot be a straw between S18+11 and S18+12. And this is different from the other "blackouts" because S18>S17.

In total we have 19 slots where there cannot be straws. There were 44 slots to begin with, so there are 44-19 = 25 slots available for the 30 straws. This contradicts the assumption that there is a solution.
[/spoiler]

I'm a bit worried that I may have missed something because the problem is not on the boundary of what is possible.