The socialite Amelia said to fellow socialite Bedelia, "I have three daughters. Can you figure out the ages of each of them knowing that the sum of their ages is 11?"
"That is not enough information," replied Bedelia.
"The product of their ages is either 16 years less or 16 years more than your age," added Amelia.
"Still not enough information." replied Bedelia after careful thought.
"My oldest daughter is learning to play chess." said Amelia.
Bedelia was then immediately able to determine the ages of Amelia's three daughters.
What are their ages?
Solution to the Problem:
The required ages of Amelia's daughters are 1,2, and 8.
The key to the problem is that Bedelia was not able to determine the ages of the daughters after she was told that the product was either 16 years more or 16 years less than her own age. That means that there must be two numbers that are the same after adding or subtracting 16 to the product of the ages.
The table below shows all the possible combinations of ages which add up to 11 for daughters #1, #2, and #3. Highlighted are the two possibilities mentioned above.
| #1 | #2 | #3 | product | product - 16 | product + 16 |
|---|---|---|---|---|---|
| 1 | 1 | 9 | 9 | -7 | 25 |
| 1 | 2 | 8 | 16 | 0 | 32 |
| 1 | 3 | 7 | 21 | 5 | 37 |
| 1 | 4 | 6 | 24 | 8 | 40 |
| 1 | 5 | 5 | 25 | 9 | 41 |
| 2 | 2 | 7 | 28 | 12 | 44 |
| 2 | 3 | 6 | 36 | 20 | 52 |
| 2 | 4 | 5 | 40 | 24 | 56 |
| 3 | 3 | 5 | 45 | 29 | 61 |
| 3 | 4 | 4 | 48 | 32 | 64 |
From the table, we can ascertain that Bedelia is 32 years old. But there are two possible ages for the three daughters:
1, 2, and 8 and
3, 4, and 4.
The clue about Amelia's oldest daughter learning to play chess tells us that there is only one oldest daughter so the solution to the ages of the daugters is 1, 2, and 8.
Correctly solved by:
| 1. Ivy Joseph | Pune, Maharashtra, India |
| 2. Braxton Simmons |
Mountain View High School, Mountain View, Wyoming, USA |
| 3. Rob Miles | Northbrook, Illinois, USA |
| 4. Kamal Lohia | Hisar, Haryana, India |
| 5. Dr. Hari Kishan |
D.N. College, Meerut, Uttar Pradesh, India |
| 6. Seth Cohen | Concord, New Hampshire, USA |
| 7. Sudhir Bavdekar | Mumbai, India |
| 8. Davit Banana | Istanbul, Turkey |
| 9. Carlos de Armas | Barcelona City, Spain |
| 10. Theta Lethe | ------------ |