## Birthday puzzle of Duorp in planet Urp with a different calendar

Duorp was born in planet Urp on the first day of the fourth month on a Fourday. Each week has 5 days, each month 10 days, a year 180 days except..read on...

### The Calendar math puzzle in planet Urp

On the planet Urp, most years have 180 days (18 months of 10 days each) except one in every seven years, when the year has 181 days, the extra day falling in the 11 day third month. Each Urp week has is five days long. The days of the week, in order of occurrence, are Oneday, Twoday, Threeday, Fourday and Fiveday. Duorp was born on a Fourday, on the first day of the fourth month in a 181 days year. On what day of the week was Duorp's 25th birthday?

### Solution thoughts

Changing suddenly the calendar we use from generations and transporting us to a whole new planet with a totally new calendar is hard to bear. But would we be too confused and stop thinking?

The truth is, **the basics of the new calendar are same:** the year, the week, the weekdays, the months, all are there, only in a different form.

Most important is the fact that the **concept of birthday (on the same date of the same month after one year) is not touched.** We can assume it as we are used to. These are in our favor.

We won't go into calendar formula and theory. Instead we would simply count days till the 25th birthday of Duorp using the information given in the puzzle.

#### Let us prepare the ground

Fortunately, Duorp was born in a 181 days year. Next 181 days year will come on the seventh year counting from the next. In the intervening six years the year will be 180 days long as usual.

It means, Duorp's 1st birthday will be in the fourth month of the next year on the first day of the month. In fact, Duorp's every birthday, including 25th birthday, will be on the fourth month of the year on the first day of the month.

Let us get down to finding the weekdays.

#### Counting weekday till the next month and the next six years

Every month has two weeks in a normal year, every week is 5 days long, every month is 10 days long and every year is of 180 days and 18 months usually . The weekdays from Duorp's birthday are:

- First month,
**birth month**(10 days):**4**5 1 2 3 4 5 1 2 3 - Second month (10 days): 4 5 1 2 3 4 5 1 2 3
- Nineteenth month (10 days):
**4 (first birthday)**5 1 2 3 4 5 1 2 3

Next month's first day will also be a Fourday. This will continue to be so till the sixth year after Duorp's birthyear. So Duorp's sixth birthday will also be a Fourday.

*This is the first breakthrough pattern.*

#### Effect of 181 days seventh year on weekdays

On the seventh year from Duorp's birthyear, the third month will have 11 days.

As the third month of Duorp's seventh birth year has 11 days, its last day that was usually a Threeday, will now be a Fourday.

The seventh birthday, the first day of the fourth month of the seventh year will then be a Fiveday, instead of a Fourday.

In other words, every seven years, the weekday of the first day of the fourth month will move ahead by one.

*This is the second breakthrough pattern.*

#### We can see the solution

On the 14th birthday, weekday will be Oneday, and on 21st birthday it will be Twoday.

On Duorp's 21th birthday, after three 7 year spans, the weekday of the first day of the fourth month will be pushed forward three times from Fourday to Fiveday to Oneday to Twoday.

And that will remain unchanged till the 25th year.

Duorp's 25th birthday will be on the first day of the fourth month on the 25th year from his birth year and **the day will be a Twoday.**

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