Brothers solving mathematical problems
A pair of brothers kept solving mathematical problems from Monday to Saturday in a certain week. The elder brother solved one problem immediately at the beginning of Monday, while the younger brother solved one at noon on Monday. It is known that the elder brother solved at least one problem during any continuous 24-hour period, while the younger brother solved at least one problem during any continuous 18-hour period. Furthermore, it was the elder brother who first solved a problem on Monday, Wednesday and Friday, while it was the younger brother who first solved a problem on Tuesday, Thursday and Saturday. What is the minimum total number of problems solved by the brothers during these six days?
The answer is 15.
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1 solution
It's pretty obvious that each brother has to solve a problem every day, since there are 24 hours in a day. So, we need to find days when brother solved more than one.
For example, if on Monday the elder brother solved only one problem, then he had to solve a problem right at the start of the next day. But on Tuesday the younger brother was the one to solve a problem first. That means, elder brother had to solve another problem on Monday.
Let's consider the possibility, that the younger brother solved just one problem on Monday and Tuesday each. That would mean he solved a problem at 12:00 on Monday, 6:00 on Tuesday and the start of the next day. But again, on Wednesday elder brother was the to solve a problem first. That means either Monday or Tuesday younger brother solved an extra problem.
Younger brother solve a problem in any 18-hour period, which means in a period of 72 hours he solves at least 4 problems. That means that in 72 hours after he solved his last problem on Tuesday younger brother had to solve 4 problems. That would there had to be an extra solved problem on one of three next days: Wednesday, Thursday, Friday.
Thus, we found that each brother solved one problem every day and there were three additional problems solved: one by elder brother on Monday, another two by younger brother on periods Mon-Tue and Wed-Thu-Fri. That makes it 6 + 6 + 1 + 2 = 1 5 problems solved.
Here's an example of possible solving schedule with only 15 problems solved.
Monday
00:00 elder brother solves a problem
12:00 younger brother solves a problem
23:00 elder brother solves a problem
Tuesday
05:30 younger brother solves a problem
22:00 elder brother solves a problem
23:00 younger brother solves a problem
Wednesday
16:00 elder brother solves a problem
16:30 younger brother solves a problem
Thursday
10:00 younger brother solves a problem
15:00 elder brother solves a problem
Friday
03:00 elder brother solves a problem
03:30 younger brother solves a problem
21:00 younger brother solves a problem
Saturday
01:00 younger brother solves a problem
02:00 elder brother solves a problem