## Saturday, 19 May 2012

### A Concept on Days and Dates

Problems are built on simple know-how things. For example, we can consider problems on Days and Dates. Here, the concept is simple and is based on simple remainder logic. Let us consider some known-facts and probable questions based on those known-facts.

What we know: “A week consists of 7-days”

Question: what is the date of next immediate Monday?
Statement-A: Today is Monday
Statement-B: Today is 3rd of Feb

The way approach is like this:
How many days are there in a week? Answer “7”. That means, the week gets repeated on every eighth day. If Day-1 is X, one cycle of the week gets completed on Day-7 and the second cycle gets started on Day-8. We get Day-8 by adding 7 days to Day-1. So, continue the logic, we get the next X-day on 8+7 = Day-15. A simple addition of 7 gives us the next date of the same-day.
If we consider the two given statements, the input is, today is Monday and 3rd Feb. Let us add a 7 to the date. 10th Feb is Monday and 17th Feb is also a Monday so on and so forth.
Obviously, the next immediate Monday is 10th Feb and the statements together are sufficient to get the answer.
Let us introduce a small twist in to this logic.

Question: what is the date of next immediate Monday?
Statement-A: Today is Monday
Statement-B: Today is 23rd Feb

How to answer this? Can we simply add 7 to 23 and tell the answer 30th Feb? No, it’s a blunder. No year can have 30 days in its February month. It’s given that, 23rd Feb of the year is Monday. It’s not given which year is that. It may be a leap-year or non-leap year. Is the given information enough to answer the question? No, definitely not.
What is this Leap-Year?
A normal year consists of 365 days and a leap-year consists of 366 days. Where from this extra day in a leap-year comes from? A normal year has 28 days in its Feb’ month where as a leap-year’s Feb’ month consists of 29 days. There it is.
What we know: ”A normal year consists of 365 days and a leap-year consists of 366 days”

Question: Is that a leap year?
Statement-A: That year starts with Tuesday
Statement-B: That year ends with Wednesday

How to find? To find whether it is a leap-year or not, it requires to find the number of days in that year. Is it possible to find it from the given conclusions? Yes, it’s possible as there are only two possibilities, 365 or 366 days.
Case for 365-days: 365/7 gives a remainder of 1 => 365 = 364+1 = (a multiple of 7) + 1
=> A period of 365 days contains one day in excess of some exact number of weeks
=> If the start-day is Tuesday, the 364th day becomes Monday such that it completes a set of weeks and the 365th day becomes a Tuesday again.
Case for 366-days: Following the logic mentioned above, for a Leap-year, if start-day is Tuesday, then the end-day becomes Tuesday+1 = Wednesday.