*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:

BornAge in 20211929 92

1938 83

1947 74

1956 65

1965 56

1974 47

1983 38

1992 29How 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 <10**then 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**