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.
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.