I have always been interested in mathematics and tend to take a mathematical view on many things in life. Recently I have been presented with a challenge that I hope yo can help me with.

A friend of mine has an excellent memory for dates and events that occurred on that date, but his party piece is the ability to tell you the day of the week that fell of any given date. This talent is all the more extraordinary because the answer comes back in less than 2 seconds (often under 1 second). By his own admission he is no mathematician.

Now his memory and knowledge are without question, but I challenged him that because of its diversity this party piece could not be based upon memory, but on mathematics. I believe that with the correct mathematical approach and the use of a common algorithm, anyone with a basic mathematical mind can do this in their head.

He agreed that is was quite easy when you know how. There lies my challenge (and now yours too!). Clearly, the 1st January would fall on the same day every 7 years if it wasn't for those annoying leap years. The principle must be based on this type of concept though.

Can you help ?

Best regards,

Neil.

Neil,

Because of irregularities in our calendar, you need a good memory and some arithmetic ability to figure out days of the week in your head. There is a formula (associated with the name of Zeller) that can be found on the web. But using that certainly takes more than a few seconds. The fastest method for most people is to memorize a sufficient number of dates to allow you to get nearby dates easily. You have to work "mod 7", which means that you reduce all numbers to their remainder after division by 7. Try looking at John Conway's method, which can be found on the web page

quasar.as.utexas.edu/BillInfo/doomsday.html

It is based on the description in his book WINNING WAYS. Conway obviously has a good memory. The web page was written for the year 1997. It refers to a 1997 calendar (for which January 1 was Wednesday, and Conway's special days 4/4, 6/6, 8/8, 10/10 and 12/12 are Friday). "This century" for him means the 1900's, but it is easy to extend his method to the 2000's (as explained on the web page). Let us know if you find the method too hard. There are easier methods, but they require even more memory.
By the way, there is a 28-year rule (whereby dates repeat every 28 years when there is no century year in the way). So, for example, March 1 was a Thursday in 1900, 1928, 1956, 1984, 2012 (because 2000 was a leap year it doesn't cause any problems), 2040, etc. The obvious rule is that the day goes forward by 1 each common year -- because a common year is composed of 52 weeks plus a day (so March 1 was Friday in 1901, Saturday in 1902, Sunday in 1903; but 1904 is a leap year with an extra day in February, so March 1 was Tuesday in 1904. Conway's method is based on the 12-year rule whereby the day goes forward by 1 every 12 years when there is no century in the way (so that March 1 1912 was Thursday + 1 = Friday). You will figure out other tricks as you get experience.

Chris