Bamboozling Brenda
One day, the Tan family went out to a shopping centre where they bumped into Mrs Tan's secondary school friend. Mrs Tan's friend was surprised to see that Mrs Tan had five children. She asked, "How old are your children?" Mrs Tan's only daughter, Brenda, answered:
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I am the middle child among the five of us.
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We have five distinct positive integers for our ages which interestingly add up to 100.
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Charles is my eldest brother. His age is a perfect square and a factor of 100.
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Darius is immediately older than me while Alfred is immediately younger than me.
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My youngest brother is Eric. My age is the geometric mean of Charles' and Eric's ages.
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Exactly one of our ages is a prime number.
Can you help Mrs Tan's friend? Give your answer as the ages of the five children from the oldest to the youngest child.
You are encouraged to use a mathematical logic approach to solve this problem. Do not be swayed by what seems apparently logical or ridiculous.
The answer is 2522201716.
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1 solution
Considering fact #3, there are only four possibilities for Charles' age - 1, 4, 25 and 100. If Charles is 1, 4 or 100 years old, then we would violate fact #2. For all five children to have distinct positive integers for their ages, Charles must be at least 5 years old. On the other hand, if Charles is 100 years old, then his four younger siblings would be 0 years old which violates fact #2. Hence Charles must be 25 years old.
Next let's look at fact #5. There are 4 possibilities for Charles', Brenda's and Eric's ages - (25, 5, 1), (25, 10, 4), (25, 15, 9) and (25, 20, 16). Consider the first possibility. The sum of their ages would be 31, which together with fact #2, tells us that Darius' and Alfred's ages add up to 100-31=69. This means that Darius must be at least 34.5 which contradicts with the fact that Charles is the oldest of the five. Similarly, for the second possibility, Darius must be at least (100-25-10-4)/2 = 30.5 while for the third, Darius must be at least (100-25-15-9)/2 = 25.5. Hence only the fourth possibility is valid. Charles is 25, Brenda is 20 while Eric is 16. Indeed, the sum of their ages would be 61 which means Darius' and Alfred's ages add up to 39.
Considering fact #6, we see that neither 25, 20 nor 16 is prime so either Darius' or Alfred's age is a prime. When we split 39 into two positive integers, we also need to bear in mind that Darius' age must be within the range of 21 to 24 and Alfred's age, within the range of 17 to 19. Considering fact #4, Darius is older than Brenda but younger than Charles while Alfred is older than Eric but younger than Brenda.
39 = 24+15 = 23+16 = 22+17 = 21+18. The first two possibilities are out as one of the ages is outside the range we just identified. 21+18 is also out as neither of them is prime. Hence Darius must be 22 while Alfred must be 17 (a prime number).
In order of decreasing age (Charles, Darius, Brenda, Alfred, Eric), our answer is 2522201716.