Calendar Peculiarities

The calendar holds many recreational ideas that can be exploited to turn one on to mathematics—or at least to explore number relationships.

Consider any calendar page, say, September 2013.

Sun	Mon	Tues	Wed	Thu	Fri	Sat
1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30

Select a (3 by 3) square of any nine dates on the calendar. For example, select those shaded above. Add 8 to the smallest number in the shaded region and then multiply by 9:

(11 + 8) x 9 = 171

Then multiply the sum of the numbers of the middle row (18+19+20=57) of this shaded matrix by 3. Surprise! It is the same as the previously arrived at answer, 171.

But why?

Here are some clues: The middle number is the mean (or average) of the nine shaded numbers. The sum of the numbers in the middle column is one-third of the sum of the nine numbers.

Now, what the probability is of 4/4 (April 4), 6/6 (June 6), 8/8 (August 8), 10/10 (October 10), and 12/12 (December 12) all falling on the same day of the week?

The probability is 1, certainty! But why this surprising result?

They are all exactly nine weeks apart. Such little known facts always draw an interest that otherwise would be untapped.