For a strong class (say a group of Grade 11s or 12s), you can generalize this problem to any birth year, so that the students can determine the day of week for their own birthday (e.g. June 15, 1978 was a Thursday). To do this, divide the participants via the “jigsaw method”, where each four-person team is assigned a letter (A, B, C, D), and the new letters form the new groups. For example, if there are 24 students in the class, the six groups of four become four groups of six.
And in these new groups, students solve the same problem, replacing the year 2019 with their birth year. In solving this harder problem, students will realize that each 365-day year contributes one extra day (52 weeks plus 1 day). Thus, if January 1, 2019 is a Tuesday, then January 1, 2018 is a Monday. In other words, when we go from 2019 to 2018, we simply subtract 1, to go from Tuesday to Monday.
Students can use the information they know (their day of week for 2019) and work backwards until they find the correct answer for their year of birth, taking leap years into account.
During one of our school visits (in 2018), one student made the powerful insight that her birthday in 2001 must be the same day of week as her birthday in 2018, since there are 17 “extra days” in addition to the four Feb 29 “leap days” that occurred in 2004, 2008, 2012, and 2016. Since 17+4=21, the calendar shifted 21 days between her birthday in 2001 and her birthday in 2018. And since 21 is a multiple of 7, if her birthday fell on a Tuesday in 2018, then it must have fallen on a Tuesday in 2001. This is how to handle the tricky concept of leap years.
And a different student observed that the calendar repeats itself every 28 years, since each year contributes one extra day (52 weeks plus 1 day), and there are 7 occurrences of February 29 during any 28-year period. Thus, the calendar shifts by 28+7 = 35 days, which is a multiple of 7.
This insight can be used to solve the Calendar Problem for adults. For example, when I shared this problem at a Professional Development workshop for teachers, one 56-year-old math teacher observed that since his birthday fell on a Wednesday, that implied that he was born on a Wednesday.
And then for super-keen students, encourage them to extend this algorithm to other famous dates in world history, dating back centuries. Some examples are June 6, 1944 (D-Day), July 1, 1867 (Confederation Day in Canada), July 4, 1776 (Independence Day in the USA), April 23, 1616 (Death of William Shakespeare), April 23, 1564 (Birthday of William Shakespeare).
Students will have to be careful about ensuring the correct calculation of leap years, due to the quirky rules that occur when the year is a multiple of 100 but not a multiple of 400. Specifically, the years 1600 and 2000 are leap years, while the years 1700, 1800, 1900 are not leap years.
For your strongest students, here is one final challenge problem:
Create your own algorithm to determine the correct day of week for any date in the 20th or 21st century (Jan 1, 1901 to Dec 31, 2100). Your algorithm should be one that you can compute in less than twenty seconds, either in your head or by jotting down some calculations on a sheet of paper. Present your algorithm, and clearly and carefully justify WHY your algorithm outputs the correct day of week, no matter what input date is chosen. Finally, verify your algorithm using the date October 29, 1929, a famous date in world history known as “Black Tuesday”.
I will purposely not post the solution to that challenge problem here. However, your students are encouraged to write out their solution, scan it as a PDF, and then e-mail it to me (richard.hoshino@gmail.com). I would be happy to write to the student and provide feedback on their solution.