A couple of days ago, I visited my relatives to celebrate my cousin's and grandmother's birthday (they share the same birthday). One was turning 18, the other 81. One of my other cousins made a nice-looking "dual birthday card," which worked because reversing the digits of "18" gave "81." Someone asked if this could happen again, and I said it would happen when my cousin turned 29 and my grandmother turned 92.

This got me to wonder what kinds of numbers this would work with, so I made a short Python script to test it. I found out this can only occur when the age difference is divisible by 9. Not only that, but it follows an interesting pattern; here's the output of my program:

Age difference 9: (1, 10), (12, 21), (23, 32), ( 34, 43), (45, 54), (56, 65), (67, 76), (78, 87), (89, 98)
Age difference 18: (2, 20), (13, 31), (24, 42), (35, 53), (46, 64), (57, 75), (68, 86), (79, 97)
Age difference 27: (3, 30), (14, 41), (25, 52), (36, 63), (47, 74), (58, 85), (69, 96)
Age difference 36: (4, 40), (15, 51), (26, 62), (37, 73), (48, 84), (59, 95)
Age difference 45: (5, 50), (16, 61), (27, 72), (38, 83), (49, 94)
Age difference 54: (6, 60), (17, 71), (28, 82), (39, 93)
Age difference 63: (7, 70), (18, 81), (29, 92)
Age difference 72: (8, 80), (19, 91)
Age difference 81: (9, 90)

I notice a couple things
1) It starts at 9, with 9 pairs. As it progresses the number of pairs decreases by one.

2) The number pairs themselves follow a pattern. The difference between the two digits is the same. e.g. for age difference 45, notice that 16, 27, 38, and 49 all have a "gap" of 5 between the two digits. The gap is (Age difference)/9.


Now, the next age difference to have pairs is 99. It has 90 pairs. Here's the pattern for age differences 99-1000:

Age difference 99: [90 pairs]
Age difference 198: [80 pairs]
Age difference 297: [70 pairs]
Age difference 396: [60 pairs]
Age difference 495: [50 pairs]
Age difference 594: [40 pairs]
Age difference 693: [30 pairs]
Age difference 792: [20 pairs]
Age difference 891 [10 pairs]

the pattern is obvious for these...

However, there are 9 other age differences with pairs between 99 - 1000:
Age difference 189: (901, 1090)
Age difference 279: (801, 1080), (911, 1190)
Age difference 369: (701, 1070), (811, 1180), (921, 1290)
Age difference 459: (601, 1060), (711, 1170), (821, 1280), (931, 1390)
... (don't really feel like typing these)

The ones with trailing zeroes don't matter as much, because for example (901, 1090) could also be (901, 10900) etc. to infinity.

All of this is when I don't test any more years after the smallest number is 1,000. If I extend that to 100,000 (which takes long to compute) there are some more numbers, such as age difference 180 which has 72 pairs starting at (1021, 1201). And then there is age difference 999, which I believe has 900 pairs.


Umm...so that's about it...this is a pretty long post of unorganized info, so I hope someone has the patience to read it.

I should *really* study number theory, because I do this kind of thing a lot with various problems and it always mystifies me...