The Mexican historical figure Benito Juárez was born in the first half of the 19th century (the 19th century spans from the year 1801 to the year 1900). Knowing that Benito Juárez turned $x$ years old in the year $x^{2}$, what was the year of his birth?

#
Solution

The perfect squares that are closest to 1801-1900 are:

$$
\begin{aligned}
& 42 \times 42=1764 \\
& 43 \times 43=1849 \\
& 44 \times 44=1936
\end{aligned}
$$

Let $x$ be the age of Benito Juárez in the year $x^{2}$. The number $x$ cannot be 42, because in this case Benito would not have been born in the 19th century (the year 1764 is not in the 19th century, which runs from the year 1801 to the year 1900). Let's now test the year 1849. If Benito's age in 1849 is 43, then Benito was born in 1849-43 = 1806. Since 1806 belongs to the 19th century, this is the correct answer. Note that it is the only possible one, because from the number 44 onwards, the square of the number minus the number itself is greater than 1901 and therefore would not belong to the 19th century.