Your digit-swapping birthday.

a mathematical riddle from my inbox

Earlier this year I received a cheerful, pleasant riddle from a cheerful, pleasant fellow named Simon.

Today, as my mother (born in 1947) turns 74, I realized that I (born 1983) will turn 38 this year. Investigating it further, it turns out that this year everybody whose sum of digits (last two digits) of year of birth is 11 gets this result. Examples:

Born         Age in 2021

1929            92
1938            83
1947            74
1956            65
1965            56
1974            47
1983            38
1992            29

How is that? I fail to find the mathematic principle that leads to this result. Would you care to help?

First: Happy belated birthday to Simon’s mother!

Second: What a nifty observation. Exactly the kind of head-scratcher I love to bring into the classroom.

Admittedly, facts about digits rarely strike at deep mathematics. After all, the patterns vanish when translated out of base ten. But who cares? They’re a fun playground for conjectures, and good practice at teasing out “funny coincidence” from “necessary consequence.”

So let me pose my own question, a riff on Simon’s: How special is this year for digit-swapping birthdays?

Spoilers follow!

Everybody will have a digit-swapping birthday. Let’s say the last two digits of your birthyear are ab (as in 19ab or 20ab or, if you are the oldest person on earth, 18ab). In that case, your digit-swapping birthday occurs when you turn ba years old.

In what year does this happen? Well, it works like this:

• Begin with the century of your birth (1900 or 2000).
• Add a decades and b years (to get to your birth year).
• Add b decades and a years (to get to your digit-swapping year).

Thus, if we define n as a+b, your digit-swapping year is [century] + n decades + n years.

If n < 10, then things are pretty simple. Your digit-swapping birthday happens in the same century as your birth, in a year of the form 19nn or 20nn.

N	Digit Swapping Year	Birth Years
1	1911	1901, 1910
2	1922	1902, 1911, 1920
3	1933	1903, 1912, 1921, 1930
4	1944	1904, 1913, 1922, 1931, 1940
5	1955	1905, 1914, 1923, 1932, 1941, 1950
6	1966	1906, 1915, 1924, 1933, 1942, 1951, 1960
7	1977	1907, 1916, 1925, 1934, 1943, 1952, 1961, 1970
8	1988	1908, 1917, 1926, 1935, 1944, 1953, 1962, 1971, 1980
9	1999	1909, 1918, 1927, 1936, 1945, 1954, 1963, 1972, 1981, 1990

But what n > 9? Then your digit-swapping birthday will land in the century after your birth, and will occur in a year of the form 20(n – 9)(n -10).

N	Digit Swapping Year	Birth Years
10	2010	1919, 1928, 1937, 1946, 1955, 1964, 1973, 1982, 1991
11	2021	1929, 1938, 1947, 1956, 1965, 1974, 1983, 1992
12	2032	1939, 1948, 1957, 1966, 1975, 1984, 1993
13	2043	1949, 1958, 1967, 1976, 1985, 1994
14	2054	1959, 1968, 1977, 1986, 1995
15	2065	1969, 1978, 1987, 1996
16	2076	1979, 1988, 1997
17	2087	1989, 1998
18	2098	1999

In each century, there are 100 birth cohorts. Their digit-swapping birthdays come in clusters: 9 pairs of years, scattered across the century, in the form 20c(c-1) and 20cc, for c = 1 to 9. In each pair of years, exactly 11 cohorts celebrate digit-swapping birthdays.

First Year	Cohorts	Second Year	Cohorts
2010	9 (turning 91, 82, 73, 64, 55, 46, 37, 28, 19)	2011	2 (turning 1, 10)
2021	8 (turning 92, 93, 74, 65, 56, 47, 38, 29)	2022	3 (turning 2, 11, 20)
2032	7 (turning 93, 84, 75, 66, 57, 48, 39)	2033	4 (turning 3, 12, 21, 30)
2043	6 (turning 94, 85, 76, 67, 58, 49)	2044	5 (turning 4, 13, 22, 31, 40)
2054	5 (turning 95, 86, 77, 68, 59)	2055	6 (turning 5, 14, 23, 32, 41, 50)
2065	4 (turning 96, 87, 78, 69)	2066	7 (turning 6, 15, 24, 33, 42, 51, 60)
2076	3 (turning 97, 88, 79)	2077	8 (turning 7, 16, 25, 34, 43, 52, 61, 70)
2087	2 (turning 98, 89)	2088	9 (turning 8, 17, 26, 35, 44, 53, 62, 71, 80)
2098	1 (turning 99)	2099	10 (turning 9, 18, 27, 36, 45, 54, 63, 72, 81, 90)

Hey, that’s only 99 cohorts. What about the final cohort?

Well, those born in 2000 don’t really get a digit-swapping birthday. Or perhaps they get two: one right away, in 2000, and another in 2100, if they can live to a century.

Finally, an answer to the riddle!

2021 has eight digit-swapping cohorts. That ties it with 2088 for third-most, trailing only the nine-cohort years of 2010 and 2099. That puts it at the 96th percentile of digit-swapping specialness. In short: this year is more special than 24 out of 25 years.

Pretty special!