Bats in a number of Caves (THE BAT CAVES!!!)

Question

This problem is kind of like a combinatorics type of problem, I believe. But also has the feeling that it kind of makes use of the so-called "pigeonhole principle". I recall doing a problem like this many years ago in University in a Discrete Mathematics course.

See problem below:

For the part of this problem, where any 7 caves in a row contains exactly 77 bats. I just did simple algebra of 7K =77, and got K = 11 bats in each cave. It satisfies the condition that there has to be at least 2 bats in every cave. But I am getting confused as to how to handle the very beginning and very last of the caves. Hope someone can provide some clarity on this.

Answer 1

the numbers in each cave must repeat with a period of 7, so cave 1 has the same number as cave 43 and cave 3 the same as cave 45.

Also caves 4 to 45 contain a total of $6 \times 77 = 462$ bats

• so there must be a total of $490 - 462 = 28$ bats in caves 1, 2 and 3.
• the minimum numbers for caves 1 and 3 are 14 and 2 respectively,
• so the maximum number in cave 2 is $28 - 16 = 12$ bats.