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Sophie hasn't been alive for a long time.
But even generations later, a curious story from 1898 is told about her again and again at family get-togethers.
That year, Sophie reached an age that exactly matched the checksum of her year of birth.
How old was Sophie then in 1898?
Sophie was 22 years old.
She was born in 1876.
The solution can be found by skillful trial and error, but then you are not necessarily sure whether this is the only possible solution.
The following solution is more elegant.
First of all, we can say that Sophie's age has an upper limit.
Because the cross sum of your year of birth cannot be arbitrarily large.
If she was born in the 19th century, 1898 is the number with the largest possible composite sum - this is 26. In the previous century, the largest possible composite number is also 26 (year 1799).
Sophie cannot have been born before 1700 - otherwise she would have been at least 198 years old.
So 26 is the upper limit for her age.
To find your specific age, let's look at the remainder when dividing by 9. It is well known that this remainder for a number is the same as the remainder of the cross sum of this number when dividing by 9. For example, the number 75 has the remainder 3 (8 * 9 + 3 = 75), the checksum of 75 is 12 and also has the remainder 3 (9 + 3 = 12).
We know that the sum of the year of birth and age is exactly 1898.
In addition, Sophie's age corresponds to the sum of the digits of her year of birth.
So we can set up the following equation:
Year of birth + cross sum (year of birth) = 1898
Now in this equation we consider the remainders when dividing by 9. These must be the same on both sides.
1898 has the remainder 8 because 1890 is divisible by 9 with no remainder.
The remainder of the year of birth and the remainder of the cross sum of the year of birth are equal.
Because the cross sum of the year of birth corresponds to Sophie's age, we can also write:
2 * remainder (age) = remainder (1898) = 8
remainder (age) = 4
This means that if you divide by 9, Sophie's age will be 4.
Because Sophie cannot be more than 26 years old, only 4, 13 and 22 years old are possible solutions.
The only possible solution is then 22 years, as we can easily verify.
This pretty riddle comes from the book "The Garden of the Sphinx" by Pierre Berloquin.
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