diamond geezer

 Wednesday, December 12, 2012


It's a very special day today. All the twelves. So special that everyone looking at today's date will say "woo, look at that", even Americans. Three numbers the same is incredibly rare, indeed so exceptional that we only get twelve such dates a century...
1/1/01, 2/2/02, 3/3/03, 4/4/04, 5/5/05, 6/6/06, 7/7/07, 8/8/08, 9/9/09, 10/10/10, 11/11/11, 12/12/12
The last three on that list are the most special of all - triplets of double-digit numbers - with 11/11/11 the ultimate perfect match. But today's your lot. You've now lived through the entire list, and you'll never see another.

Almost as very-special are dates consisting solely of identical digits. We had three of these last year, but again such dates are very rare. There are only a dozen in every century, with the next almost a decade hence.
1/1/11, 11/1/11, 1/11/11, 2/2/22, 22/2/22, 3/3/33, 4/4/44, 5/5/55, 6/6/66, 7/7/77, 8/8/88, 9/9/99
If the twenty-four dates above form the calendar's Premiership, then simple counting sequences are the Championship. There are twenty-three such triplets, eleven ascending and twelve descending.
1/2/03, 2/3/04, 3/4/05, 4/5/06, 5/6/07, 6/7/08, 7/8/09, 8/9/10, 9/10/11, 10/11/12, 11/12/13
2/1/00, 3/2/01, 4/3/02, 5/4/03, 6/5/04, 7/6/05, 8/7/06, 9/8/07, 10/9/08, 11/10/09, 12/11/10, 13/12/11
These are in-your-face enough for most people to notice, and are usually the signal for a flurry of tweets saying "woo, it's precisely 07:08:09 10/11/12" or the like. But again they're really quite rare. And, with the exception of one final straggler in 52 weeks time, they too have all passed by.

If considering digits rather than numbers, however, then there are more ascending and descending sequences to celebrate. Not many more, but fourteen still to look forward to.
1/2/34, 2/3/45, 12/3/45, 3/4/56, 23/4/56, 4/5/67, 5/6/78, 6/7/89
3/2/10, 4/3/21, 5/4/32, 6/5/43, 7/6/54, 8/7/65, 9/8/76
And then there are several additional dates of arithmetical interest. Precisely additional sometimes, for that category of dates where the day plus the month equals the year. Take for example the first day of last month, which was 1/11/12. You might not have got excited about it, but 1+11=12 was good enough to elevate this particular date into the special addition category. There were eleven such dates this year, one in every month except December, and there'll be twelve more every year from 2013 to 2030. Here are the 'addition' dates for 2013...
12/1/13, 11/2/13, 10/3/13, 9/4/13, 8/5/13, 7/6/13, 6/7/13, 5/8/13, 4/9/13, 3/10/13, 2/11/13, 1/12/13 (and 353 other dates this century)
It's no coincidence that there are 365 addition dates altogether. Every single day-plus-month combination adds to give a possible year, with the exception of Leap Day 29th February where 29/2/31 doesn't exist. Less than 20% of the 21st century's addition dates have passed, so we still have plenty more of these to enjoy.

Let's move on to subtraction. Every year from 2000 to 2019 sees twelve subtractable dates, one per month, then they gradually fade away. From the first of the century on 1/1/00 to the last on 31/1/30, there are 299 of these in total. Here, for example, are all the 'subtraction' dates for 2013...
14/1/13, 15/2/13, 16/3/13, 17/4/13, 18/5/13, 19/6/13, 20/7/13, 21/8/13, 22/9/13, 23/10/13, 24/11/13, 25/12/13
Americans with their month/day order saw no subtraction dates in 2012, indeed will see no more this century. But we've got another on Christmas Eve, because twenty-four subtract twelve is twelve, and then a further 143 to go.

My favourites are the multiplicative dates. They're a bit more subtle, a bit less obvious, so they make me smile even if most people never notice them. A simple times table like 6/2/12 might be clear enough, but who'd spot 26/3/78 or 13/7/91 unless they had a particular mathematical bent? There are 212 such dates altogether, a little better spread out than all the other dates we've considered so far. Here, for example, are all the 'multiplication' dates for the second half of the century...
25/2/50, 10/5/50, 5/10/50, 17/3/51, 26/2/52, 13/4/52, 27/2/54, 18/3/54, 9/6/54, 6/9/54, 11/5/55, 5/11/55, 28/2/56, 14/4/56, 8/7/56, 7/8/56, 19/3/57, 20/3/60, 15/4/60, 12/5/60, 10/6/60, 6/10/60, 5/12/60, 21/3/63, 9/7/63, 7/9/63, 16/4/64, 8/8/64, 13/5/65, 22/3/66, 11/6/66, 6/11/66, 17/4/68, 23/3/69, 14/5/70, 10/7/70, 7/10/70, 24/3/72, 18/4/72, 12/6/72, 9/8/72, 8/9/72, 6/12/72, 25/3/75, 15/5/75, 19/4/76, 11/7/77, 7/11/77, 26/3/78, 13/6/78, 20/4/80, 16/5/80, 10/8/80, 8/10/80, 27/3/81, 9/9/81, 28/3/84, 21/4/84, 14/6/84, 12/7/84, 7/12/84, 17/5/85, 29/3/87, 22/4/88, 11/8/88, 8/11/88, 30/3/90, 18/5/90, 15/6/90, 10/9/90, 9/10/90, 13/7/91, 23/4/92, 31/3/93, 19/5/95, 24/4/96, 16/6/96, 12/8/96, 8/12/96, 14/7/98, 11/9/99, 9/11/99
Some years do better than others for multiplication. 2060 and 2072 appear in the list six times (as do 2012, 2030, 2036 and 2048), but the best year for multiplicative dates is 2024 with seven. Only twenty-one years have none at all, and it'll be 2037 before we reach the first of those.

At the bottom of the arithmetical ranking comes division. These dates aren't quite so exciting, nor indeed as common, because dividing a day by a month generally leaves a remainder. Many are boring divisions with a 1 in them, like 12/1/12, which ought to be excluded because they've already appeared in the multiplication list. But there are 58 other dates unique to division, of which only 26/2/13 and 28/2/14 have yet to occur.

Here's how many special dates of each type I think there are in a century (being careful not to double-count any repeats when calculating the total).

type of special datefirst and lastin a century
equal numbers1/1/01 to 12/12/1212
repeated digits1/1/11 to 9/9/9912
ascending sequence1/2/03 to 11/12/1311
descending sequence 2/1/00 to 13/12/1112
ascending digits1/2/34 to 6/7/898
descending digits3/2/10 to 9/8/767
addition1/1/02 to 31/12/43365
subtraction1/1/00 to 31/1/30299
multiplication1/1/01 to 9/11/99212
division2/2/01 to 28/2/1458

So, if my calculations are correct, every century contains a total of 993 special calendar dates. That means a day picked at random has about a 2½% chance of being special, according to my subjective definition of what special is.

And which is the most special year of the century for special dates? Why, it's 2012, with 33 altogether.
11/1/12, 12/1/12, 13/1/12, 6/2/12, 10/2/12, 14/2/12, 24/2/12, 4/3/12, 9/3/12, 15/3/12, 3/4/12, 8/4/12, 16/4/12, 7/5/12, 17/5/12, 2/6/12, 6/6/12, 18/6/12, 5/7/12, 19/7/12, 4/8/12, 20/8/12, 3/9/12, 21/9/12, 2/10/12, 22/10/12, 1/11/12, 10/11/12, 23/11/12, 1/12/12, 12/12/12, 24/12/12
Eleven additions, twelve subtractions, six multiplications, a division, one ascending sequence, and today's glorious triple. No other year can beat that.

In this, the calendar's most special year of the century, today is the most special day of the year. Twelve twelve twelve. Enjoy this freakish repeat while you can, because you'll never see its like again.

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