Age Whose Square Is the Year
A man once noticed something remarkable: the square of his current age equaled the current year. He wrote this down in a journal, but the ink was smudged and the entry is now unreadable.
You know two things: - He died in 1970. - He did not live longer than 100 years.
What was his age when he made the journal entry?
Hints
- The year of the entry equals his age squared. What range of years is possible given the constraints?
- He was born no earlier than 1870 and the year $x^2$ must be at most 1970. So $x^2 \in [1870, 1970]$.
- Check $\sqrt{1870} \approx 43.2$ and $\sqrt{1970} \approx 44.4$ -- the only integer in that range is $x = 44$, giving $44^2 = 1936$.
Worked Solution
How to Think About It: We need an integer age $x$ such that $x^2$ is a valid year during this man's lifetime. He died in 1970 and lived at most 100 years, so he was born no earlier than 1870. That means $x^2$ must fall in the range $[1870, 1970]$. We just need to check which perfect squares land in that interval.
Quick Estimate: $\sqrt{1870} \approx 43.2$ and $\sqrt{1970} \approx 44.4$. So the only integer in that range is $x = 44$, since $44^2 = 1936$. Quick sanity check: if he was 44 in 1936, he was born in 1892, which is between 1870 and 1970, and he would have been 78 at death in 1970 -- well under 100. Works.
Approach: Systematically check nearby perfect squares.
Formal Solution:
Let $x$ be his age when he made the entry, so the year of the entry is $x^2$.
- He died in 1970, so $x^2 \leq 1970$.
- He lived at most 100 years, so he was born no earlier than 1870, meaning $x^2 - x \geq 1870$ (birth year = year of entry minus age).
- Also $x^2 \geq 1870$ (the entry must be within his lifetime).
Check the candidates:
| $x$ | $x^2$ | Birth year ($x^2 - x$) | Age at death ($1970 - x^2 + x$) | Valid? | |-----|--------|------------------------|----------------------------------|--------| | 43 | 1849 | 1806 | 164 | No (before 1870, age > 100) | | 44 | 1936 | 1892 | 78 | Yes | | 45 | 2025 | 1980 | -- | No ($x^2 > 1970$) |
Only $x = 44$ satisfies all constraints.
Answer: His age was $44$ (the year was 1936, he was born in 1892, and he died at age 78 in 1970).
Intuition
This is a constraint-satisfaction puzzle disguised as a word problem. The key move is translating the narrative into a mathematical constraint: $x^2 \in [1870, 1970]$. Once you do that, the answer is immediate because there is only one perfect square in that 100-year window. The broader lesson: when a puzzle gives you a "remarkable coincidence" (age squared equals year), your first step should be to pin down the feasible range. Perfect squares grow quadratically while the constraint window is linear, so there is typically at most one solution.