Some friends are at a café. One tells them about a curious coincidence involving his age and his siblings’ ages. Gianni says it happened when he was 30; Fabrizio says it happened when he was 42. It never happened to Aldo. What are the age gaps, and what explains the difference?
The story and the questions
The age gap between my younger sister and me and the age gap between my older brother and me meant that, on one occasion, my age was the product of those gaps, while their ages were respectively the squares of their age gaps from me.
Aldo says, “I have a younger brother and an older brother, but it has never happened to me.”
Gianni says, “It happened to me when I was 30.”
Fabrizio says, “It happened to me when I was 42.”
- What are the age gaps between Gianni and his siblings?
- What are the age gaps between Fabrizio and his siblings?
- Why has it never happened to Aldo?
Translating the situation into algebra
Let p be the age gap from the younger sibling and g the age gap from the older brother. Both gaps are positive. At the stated moment, the protagonist’s age is the product p · g, while the other two ages are the squares of their respective gaps:
| Younger sibling | Protagonist | Older brother |
|---|---|---|
p² | p · g | g² |
Because p is the gap between the protagonist and the younger sibling, the protagonist is also p² + p years old. Because g is the gap between the older brother and the protagonist, the protagonist is also g² - g years old. These expressions must be equal:
p² + p = g² - g
p + g = g² - p²
p + g = (g - p)(g + p)Since p + g > 0, divide both sides by p + g:
1 = g - p
g = p + 1The situation can occur only when the two age gaps are consecutive numbers.
After substituting g = p + 1, the age table becomes:
| Younger sibling | Protagonist | Older brother |
|---|---|---|
p² | p(p + 1) | (p + 1)² |
Gianni’s case: age 30
For Gianni, set the protagonist’s age equal to 30:
p(p + 1) = 30
p² + p - 30 = 0
p = (-1 ± √(1 + 120)) / 2
= (-1 ± 11) / 2Take the positive solution: p = 5. Therefore g = p + 1 = 6.
| Gianni’s younger sibling | Gianni | Gianni’s older brother |
|---|---|---|
5² = 25 | 5 · 6 = 30 | 6² = 36 |
The age gaps are therefore 5 years from the younger sibling and 6 years from the older sibling.
Fabrizio’s case: age 42
For Fabrizio:
p(p + 1) = 42
p² + p - 42 = 0
p = (-1 ± √(1 + 168)) / 2
= (-1 ± 13) / 2The positive solution is p = 6, so g = 7.
| Fabrizio’s younger sibling | Fabrizio | Fabrizio’s older brother |
|---|---|---|
6² = 36 | 6 · 7 = 42 | 7² = 49 |
The age gaps are therefore 6 years from the younger sibling and 7 years from the older sibling.
Why has it never happened to Aldo?
The story gives neither Aldo’s age gaps nor his current age, so the available information does not tell us why he says that it has never happened. If g - p ≠ 1, the gaps are not consecutive and the event is impossible. If instead g = p + 1, the event occurs when Aldo reaches age p(p + 1), so it may still lie in his future. In that case, “it has never happened” means only “it has not happened yet.”
Prepared by Salvatore Mosaico.