## Math Explorer's Club Tips for Mental Computations

Introduction
Lesson 2: Subtraction
Lesson 3: Multiplication
Lesson 4: Division
Lesson 5: Calendar computations
Lesson 6: Guessing a number and other tricks

## Calendar computations

This is a very practical skill. Certainly anyone can appreciate someone who can tell them the day of the week they were born.

We describe a procedure to obtain the day of any date in the 21st century. A key idea is modular arithmetic. We first associate the integers module 7 to the days of the week.

 Day Code Monday 1 Tuesday 2 Wednesday 3 Thursday 4 Friday 5 Saturday 6 Sunday 0

The above codes are very easy to remember. For instance, Monday is the first (one) day of the week you have to go to school, and Tuesday (2-sday) is the second day. You can have fun making up ways to remember these codes.

We now use the following codes for the months of the year.

 Month Code January 5 in a leap year, and 6 otherwise February 1 in a leap year, and 2 otherwise March 2 April 5 May 0 June 3 July 5 August 1 September 4 October 6 November 2 December 4

It may be easier to remember if we assign words to their respective codes. For instance

January: First full month of Winter (Winter has 6 letters).
May: Think of 5 de Mayo, and think of Mayo as May-0.
August starts with "A" (A is the first letter in the alphabet).
Septermber: Fall starts in September (Fall has 4 letters).
November: Elections on Tuesday (2-sday).

Have fun and make up your own way of remembering the month codes!

Finally we assign codes for the years of the 21st century (and the year 2000). We present the codes from 2000 to 2010, but any code can be constructed from the previous one by adding one in a non-leap year and 2 in a leap year. This addition is done module 7.

 Year Code 2000 0 2001 1 2002 2 2003 3 2004 5 2005 6 2006 0 2007 1 2008 3 2009 4 2010 5

Activity: Completing the above table
1. The table above with year codes is very easy to construct. Simply add a one if for non-leap years and two for leap years.
2. Give a function (formula) f so that for a year x = 2000, 2001, . . . , 2099, f(x) gives the x's code.
3. How can we change the table to apply to the 20th century? What about the 22nd?

Now to obtain the day of a date, simply get the value module 7 of the sum Month Code + Day Code + Year Code. For instance, let's see how to figure out the day of March 3, 2010: March's code is 2, and 2010's code is 5 . The sum is

2 + 3 + 5 &equiv 3 (module 7).

So March 3, 2010 is a Wednesday.

Activity: What are the following dates?
1. July 4, 2050.
2. November 10, 2003.
3. December 2, 2098.

# Why does the above method work?

Here are some observations that are useful to understand why the above process works.

1. Modular arithmetic. For more information about this topic, we refer to another Math Explorer's club web page. We use arithmetic module 7, since a week has 7 days.
2. A year has either 365 or 366 days. Since 364 is a multiple of seven, a given date will change from one day (say from Tuesday to Wednesday) if the year has 365 days. On the other hand if the year has 366 days, then a date will change two days (say from Tuesday to Thursday). That is why the table with the year codes makes sense.

In the Activity below, we'll try to make sense of the codes given to the months of the year.

Activity: Making sense of the Month codes.
Find a calendar for the year 2000. Knowing the year codes and looking at the 2000 calendar, argue why the month codes must be the ones presented

# January 1st of the current century

What day of the week is January 1, 2098? It turns out that there is an easy way to tell the day of the week of January 1 of any year of this century. The process is as follows.

1. For a non-leap year x.
• x has the form 20ab. Take the ab and divide it by 4 to obtain y.
• Take the integer part of y.
• The value of the sum ab + y module 7 gives the day.
2. For a leap year, simply add 1 to outcome of the previous procedure.

As an example, what day of the week is January 1st, 2050?

50 divided by 4 is 12.5, that has integer part 12. Now 50 + 12 = 62 &equiv 6 (mod 7). So January 1st, 2050 is Saturday.

Activity: January 1, 20xy.
1. Why does the above method work? Show the validity of the above method by reducing it to a special case of the general method.
2. What day of the week is January 1 of
1. 2006
2. 2012
3. 2051
Cornell University, Department of Mathematics