a math puzzle from my inbox

Earlier this year I received a happy, pleasant puzzle from a happy, personable guy named Simon.

Today, as my mother (born in 1947) will be 74 years old, it became clear to me that I (born in 1983) will be 38 this year. If you examine it further, it turns out that this year anyone whose digit sum (the last two digits) of the year of birth is 11 will get 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 cannot find the mathematical principle that leads to this result. Would you like to help?

First: Congratulations on the belated birthday to Simon’s mother!

Second, what a nice observation. Exactly the kind of head scratch I like to bring into the classroom.

Granted, facts about digits rarely meet deep math. Eventually the patterns disappear when translated out of base 10. But who cares? They are a fun playground for guesswork and good practice in separating “funny coincidences” from “necessary consequences”.

So let me ask my own question, a riff on Simons: How special is this year for number swap birthdays?

Spoilers will follow!

Everyone will have a digit swap birthday. Let’s say the last two digits are your year of birth from (as in 19ab or 20ab or, if you are the oldest person in the world, 18ab). In this case, your digit swap birthday occurs when you turn around ba Year old.

What year does this happen? Well it works like this:

  • Start with the century of your birth (1900 or 2000).
  • Add a Decades and b Years (to get to your year of birth).
  • Add b Decades and a Years (to get to your digit exchange year).

So when we define No how a + b, is your digit swapping year [century] + n decades + n years.

If n <10then things are pretty simple. Your birthday to swap digits falls in the same century as your birth, in a year of form 19nn or 20nn.

No Year of digit exchange Years of birth
1 1911 1901, 1910
2 1922 1902, 1911, 1920
3 1933 1903, 1912, 1921, 1930
4th 1944 1904, 1913, 1922, 1931, 1940
5 1955 1905, 1914, 1923, 1932, 1941, 1950
6th 1966 1906, 1915, 1924, 1933, 1942, 1951, 1960
7th 1977 1907, 1916, 1925, 1934, 1943, 1952, 1961, 1970
8th 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 swap birthday will land in the century after you were born and will take place in a year of the form 20 (n – 9) (n -10).

No Year of digit exchange Years of birth
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
13th 2043 1949, 1958, 1967, 1976, 1985, 1994
14th 2054 1959, 1968, 1977, 1986, 1995
fifteen 2065 1969, 1978, 1987, 1996
16 2076 1979, 1988, 1997
17th 2087 1989, 1998
18th 2098 1999

There are 100 birth cohorts in each century. Your number swap birthdays come in groups: 9 year pairs, scattered over the century, in the form 20c (c-1) and 20cc, for c = 1 to 9. In each year pair, exactly 11 cohorts celebrate number swap birthdays.

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

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

Well, those born in 2000 don’t have an actual birthday. Or maybe they’ll get two: one immediately in 2000 and another in 2100 when they can turn a century.

Finally a solution to the riddle!

2021 has eight digit swapping cohorts. This puts it in third place with 2088, only after the nine cohort years 2010 and 2099. This puts it in the 96th percentile of the digit swapping specialty. In short: this year is more special than 24 out of 25 years.

Pretty special!

Released

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