1. ## Choosing problem

A lumberjack cuts down at least one tree everyday. He cuts down exactly 12 trees every week. Prove that we can choose some consecutive days when he cut down exactly 20 trees (in all).

Any help would be appreciated.

2. Originally Posted by doug
A lumberjack cuts down at least one tree everyday. He cuts down exactly 12 trees every week. Prove that we can choose some consecutive days when he cut down exactly 20 trees (in all).

Any help would be appreciated.
Is this what you're looking for?:

1st week - 7 trees (one a day)
2nd week - 7 trees (one a day)
3rd week - 6 trees first six days (one a day)

total is 20 trees cut down.

3. Originally Posted by wonderboy1953
Is this what you're looking for?:

1st week - 7 trees (one a day)
2nd week - 7 trees (one a day)
3rd week - 6 trees first six days (one a day)

total is 20 trees cut down.
Not really, because he cuts down exactly 12 trees every week (and not 7).
(exactly 12 trees - every week and minimum 1 - every day).

4. Examine three consecutive weeks (21 days). In this 21-day-long period the lumberjack cut off exactly 36 trees. Considering the fact that he cut off at least one tree every day and we examine how many trees he has cut till the kth day we can say that we have 21 different numbers (between 1 and 36). Among these numbers there are two numbers that are congruent modulo 20 (using pigeonhole principle). Between these two days he cut off exactly 20 trees.
I hope this prove is clear enough and I could help you.

5. And the 36 maximum makes it impossible for the difference to be 40, not 20.