Pigeonhole principle
- Thread starter Nube
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Yes, I have many ideas, but, as my wife will explain with examples, most are very bad.
Still, I might start by redefining the problem. Call a ”week” five consecutive days. Now the problem is that over six consecutive weeks, he played exactly 45 games in total, and he played at least 5 games each week. Now show that there must have been at least one week when he played exactly 14 games.
EDIT: What was the complete and exact wording of the problem?
For weeks one through five, he played five games each week. Therefore he played twenty in week six.
[imath]20 \ne 14.[/imath]
Still, I might start by redefining the problem. Call a ”week” five consecutive days. Now the problem is that over six consecutive weeks, he played exactly 45 games in total, and he played at least 5 games each week. Now show that there must have been at least one week when he played exactly 14 games.
EDIT: What was the complete and exact wording of the problem?
For weeks one through five, he played five games each week. Therefore he played twenty in week six.
[imath]20 \ne 14.[/imath]
Last edited:
It's false, right?
Dr.Peterson
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My first concern about the problem is, are "practice", "study", and "train" meant to be synonyms? I suppose so, but this isn't written carefully.
The 5 consecutive days don't have to be one of the "weeks" you break it into; they could span parts of two weeks.
In your example, as I read it, he played one game each of the first 25 days, and perhaps 4 on each of the last 5 days. So one set of 5 consecutive days would be 1, 1, 4, 4, 4 games, which is exactly 14.
On the other hand, if it's 1 game on each of the first 29 days, and 16 on the last day, then we see the claim is false. The sum for any five days in this case is always either 5 or 20, never 14. (In my mind, thinking about counterexamples is a central part of the pigeonhole concept: What would it take for the claim to be false?)
Again, we need to see the exact wording of the problem, and particularly whether it asks you to prove it true, or to decide whether it is true and prove that.
blamocur
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I believe it is: if he practiced 2 times/day on odd days and 1 time/day on even days then any stretch of 5 consecutive days will have 7 or 8 practices.
Yes, it’s obviously false under my simplification, but Dr. Peterson has pointed out that my simplification is not a useful one because the five consecutive days need not fall in any of what I proposed as a “week.” The five consecutive days could be split across TWO weeks.
The fact remains that, as you have described the problem, there seems to be nothing that precludes playing 8 games on day 1, 7 games on day 2, 1 game on day 3, 1 game on day 4, 1 game on day 5, and 1 game on day 6, for a total of 19. That represents 18 rather than 14 on the first five days and 11 rather than 14 on the second five days. There are 26 games left to play over 24 days. Given that at least one game is played each day, no more than 3 games can be played on a single day and
1 + 1 + 1 + 1 + 3 = 7 = 1 + 1 + 1 + 2 + 2.
What is the original wording of the problem?
Steven G
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He played for sure 30 games in 30 days. But there are 15 extra games to be played in the 30 day period as well.
To me the problem becomes:
Magnus Carlsen practiced chess during a 30 day period, and in
those 30 days he studied exactly 15 times in total. Prove that there are 5 consecutive days in which
Magnus trained exactly 9 times.
To me the problem becomes:
Magnus Carlsen practiced chess during a 30 day period, and in
those 30 days he studied exactly 15 times in total. Prove that there are 5 consecutive days in which
Magnus trained exactly 9 times.