On her birthday in 2021, Anna reached an age in years equal to the sum of the digits of her birth year. In addition, Anna was less than 100 years old in 2021. In what years could she have been born? (This problem was inspired by a post by puzzle writer Presh Talwalkar.)
Anna’s birth year is in either the 1900s or 2000s. Let us first consider the first possibility. Because in 2021 her age in years was equal to the sum of the digits of her year of birth, 2021 – (1900 + 10A + B) = 1 + 9 + A + B, which can be summarized as 11A + 2B = 111. B is a digit in a year, so it has to be 9 or less. Because 2B can be at most 2 x 9 = 18, 11A ≥ 111 – 18 and A ≥ 93/11, or 8.45 as a decimal. Therefore, A = 9, which means that B = 6. In the second possibility, the equation 2021 – (2000 + 10C + D) = 2 + 0 + C + D, or 11C + 2D = 19, applies. Because 19 is odd and 2D is even, 11C, and therefore C, must be odd. Consequently, C must = 1, which means that D = 4. Anna was born either in 1996 or 2014.
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This puzzle originally appeared in Spektrum der Wissenschaft and was reproduced with permission.