D o u b l e

When a double-digit birthday occurs in a double-digit year On my 33rd birthday, a friend observed that it was a double-digit year and my age was a double-digit number. She noted that the same was true of my previous two double-digit birthdays and would be true of the next two, but not thereafter. A quick calculation reveals that anyone whose birth occurs in a double-digit year will have double-digit birthdays in double- digit years for the rest of the century. The maximum number of such birthdays one can have is nine, if '00 is the birth year. After the turn of the century there are no more, but people can start having double-digit birthdays in double-digit years of the next century if they were born in years like '89, '78, '67, '56, '45, '34, '23, '12, or '01. One-digit dates & Calendromes May 5th, 1955 was a date that could be written using only a single digit: 5/5/55. (This assumes a dating convention that puts the year-of-century last as a two-digit number, but does not depend on the order of month and year.) In most decades, there is only one of these, separated by eleven years, one month, and a day, for example 2/2/22, 3/3/33, 4/4/44, 5/5/55, 6/6/66, 7/7/77, 8/8/88, and 9/9/99. However, in the first decade of a century, there are none -- because zero is required in the year but cannot stand alone as a month or day. While there are no single-digit dates in the new century thru 31 December '10, the very next day is one (1/1/11) and there are in the same year: 1/11/11, 11/1/11, and 11/11/11. The dates 10/10/10 and 12/12/12 are interesting for their symmetry (as is 1/2/12), but these cannot be written with just one digit. Starting with 2/2/22, there are 8 more single-digit dates left in the century, occuring exactly one year, one month, and one day apart. If time of day is included, using the common digit clock notation, the most redundant time of the century is 11/11/11 at 11:11 (or 11:11:11, if we include the seconds). Thus, twelve ones may be used to write this date and time on Verterans'day ++++ Even with a 24-hour clock, there is no other date and time combnation which uses twelve of the same digit. Palindromic years occur once per century: 1991, 1881, 1771, etc. thru 1111, 1001, and back thru 999, 888, etc. all the way to 111. Within the same millenium, the spacing is 110 years, but there are only 11 years between the last one, 1991, and the next one, palindromic year 2002. Then come 2112, 2222, 2332, etc., 110 years apart again. On a standard, 12-hour digital clock, there are three truly palidromic times with four digits: 10:01, 11:11, and 12:21, and five with three digits (if we ignore the colon when looking for palindromes): 1:11, 2:22, 3:33, 4:44, and 5:55. For customary 24-hour clocks, with leading zeroes, besides the above three four-digit times (10:01, 11:11, and 12:12), there are thirteen more: six in the early hours (00:00, 01:10, 02:20, 03:30, 04:40, and 05:50), three more in the afternoon (13:31, 14:41, 15:51), and three at night (20:02, 21:12, 22:22, and 23:32). One-digit times (i.e. which can be displayed using only a single digit) occur for only six of the 1440 possible minutes that can be displayed on a standard 12-hour clock: 1:11, 2:22, 3:33, 4:44, 5:55, and 11:11. If seconds are shown, then this becomes six of the 86,400 possible seconds: 1:11:11, etc. On a standard 24-hour clock, where the hour is always shown as two digits, there are only three single-digit times: 00:00, 11:11, and 22:22 (or 00:00:00, 11:11:11, 22:22:22, if seconds are displayed). TWO-DIGIT TIMES: STAIR-STEP TIMES On an ordinary digital clock, the time 12:34 is the only time that can be written with four digits, each one-greater than the preceeding digit, is: 12:34 With seconds displayed, this magical time becomes even more profound: 12:34:56 For 24-hour clocks, the same is true and we can add 23:45, but not if seconds are displayed. On a 24-hour clock with leading zeroes, there is yet another stairstep time: 01:23 or: 01:23:45 There are four three-digit stairstep times (but on 12-hour clocks only). These are: 1:23, 2:34, 3:45, and 4:56. With seconds displayed, only the first two of these are acceptable: 1:23:45 and 2:34:56. If the step increases by two, there are three double-stepping times for a 12-hour clock: 1:35, 2:46, and 3:57, and two for a 24-hour clock: 02:46 and 13:57. Two triple-step times of 1:47 and 2:58, and one quadruple-step time of 1:59, are possible on a 12-hour clock, but there are none on a 24-hour clock. None of these is valid fro a clock which displays seconds. For a 12-hour clock, the six descending-step times are: 2:10, 3:21, 4:32, 5:43, and 6:54. Among descending times, there are four with double-steps (7:53, 6:42, 5:31, and 4:20), three with triple steps (8:52, 7:41, 6:30), and two with quadruple-steps (9:51, 8:40). With seconds displayed, only three of these is valid: 4:32:10, 5:43:21, and 6:54:32. No descending times are possible on a 24-hour clock, since the leading digit must be 0, 1, or 2.