From Jason Armstrong comes a curious conundrum of calendars:
Two friends of Jason, who happen to be married to each other, have birthdays on Feb. 9 and Nov. 18. When written numerically in MM/DD formatting, these dates are 02/09 and 11/18. Jason noted that the latter date includes both the sum and the product of the values in the former date. In other words, 11 = 02 + 09 and 18 = 02 × 09.
How many pairs of dates are there such that one of the dates includes both the product and the sum of the values in the other date (in either order)? Also, note that the order of the dates in the pair doesn’t matter, so “02/09 and 11/18” should be considered the same as “11/18 and 02/09.”
There are two dates. Lets loosely define them as the Original and Computed.
The maximum either of the numbers in the Original Date can be is 12. If there were any number greater than 12, then both the product and sum of the Computed Date would be greater than 13.
The two numbers are interchangeable unless they are equal. 01/01 is an example.
The maximum of one number in the Computed Date is 31. The maximum of the other number in the Computed Date is 12.
The two numbers are interchangeable if they are both 12 or less. However they are not interchangeable in the special case of 04/04.
Even with correct calculations, the dates could possibly not exist in the calendar.
Fortunately these never occured. It is not possible to get a product of 29 or 30 with a sum of 2. A product of 31 also never occurs.
The table below summarizes all possible pairs of dates. The x and y axis denote the numbers to create the Original Date. The Computed Date is listed at each Original Date in the plot.
Note "06/02 -> 08/12" corresponds to:
There are $3\cdot(1) + 16\cdot(2) + 15\cdot(4) = 95$ possible pairs of dates.