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A field worker has to make altogether 43 visits, at least one on each day. Is there a period of consecutive days on which he makes exactly 21 visits if he makes his visits on 22 days? What happens to the problem if he makes his visits on 23 days instead of 22 days?

How can I approach to solve this problem using Pigeonhole principle, thanks.

Yanko
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2468
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  • Variations of this question have been asked and answered on this website many times. I'll see if I can find them.... https://math.stackexchange.com/questions/97397/combinatorics-pigeonhole-principle-question and https://math.stackexchange.com/questions/571937/combinatorics-pigeonhole-problem and https://math.stackexchange.com/questions/1145254/induction-or-pigeonhole-principle-or-what and https://math.stackexchange.com/questions/1271126/pigeonhole-principle-proof-in-combinatorics – Gerry Myerson Nov 26 '18 at 11:48
  • Also https://math.stackexchange.com/questions/1636571/tricky-pigeonhole-principle-question and https://math.stackexchange.com/questions/1853532/pigeonhole-principle-question-jessica-the-combinatorics-student and https://math.stackexchange.com/questions/2163546/counting-problem-probably-related-with-pigeonhole-principle and https://math.stackexchange.com/questions/2413861/how-to-prove-that-there-is-a-string-of-consecutive-days-in-which-a-factory-produ and https://math.stackexchange.com/questions/2512137/using-pigeonhole-to-prove-disprove-a-certain-number-of-consecutive-vists and many more. – Gerry Myerson Nov 26 '18 at 11:52
  • Are you sure that your figures are right, because I'm trying to solve it and the result is not coming as expected. – Sauhard Sharma Nov 26 '18 at 12:00
  • Actually, the given figures are right and I didn't find similar questions that enable me to solve the problem and that is why I post it here, thank you. – 2468 Nov 26 '18 at 12:55

1 Answers1

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If he makes his visits on 22 days, there need not be a period of consecutive days on which he makes exactly 21 visits. He could make one visit each day for the first 20 days, then 22 visits on the 21st day, and one visit on the 22nd day. That's $20+22+1=43$ visits, at least one each day for 22 days, and clearly no set of consecutive days with exactly 21 visits.

For the 23-day version, I'd suggest having another look at the links I posted in the comments, as I am confident the methods used in those links will cover this case.

Gerry Myerson
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