Susan is so fond of Friday, the day of arrival of her favorite magazine, New Scientist, that she writes the date of every Friday of the year on a sheet of paper--just the ordinal date, so that, for example, the sheet for 2006 starts out: 6th, 13th, 20th, 27th, 3rd, ... .

She keeps these sheets in boxes by decade: one is labeled "the fifties"; another, "the sixties"; and so forth, through one she has labeled "the noughties".

She recently opened one of the earlier boxes and discovered the sheets were in tatters. Not only that, but the digits have all faded away. On one scrap, she found "rd nd" as consecutive entries. (The ordinal suffixes were presumably written in a hardier ink.) This did not enable her to work out precisely which dates these represented. But she did find within the same box, another, similar scrap showing two consecutive entries and was able to work out the precise dates to which they referred. What were those dates (including the year)?

rd followed by nd can only happen between feb 23rd and mar 2nd which can only be in a non-leap year. Now, to find any other two entries and be able to uniquely determine the date and year from them, they must be unique within the decade. rd st is the pattern, as it can only occur between feb 23rd and mar 1st in a leap year and two consecutive leap years cannot start on the same day of the week and a single decade cannot have three leap years. So we need to find a decade that has two Friday feb 23rds, one of which is in a leap year and one of which isn't. Inspection yields 1962 and 1968 as such a pair (1968 is the leap year). So the first scrap referred to feb 23rd, 1962 and mar 2nd 1962 while the second scrap, which read "rd st" referred to feb 23rd, 1968 and mar 1st, 1968.

Posted by Paul on 2007-09-01 19:37:24