Three people at a party discover that the product of any two of their ages is equal to the year in which the third was born.

What is the most recent year the party could have taken place?

If x is the current year, a is the birth year of person 1, b is the birth year of person 2, and c is the birth year of person 3, then we are trying to find:
(x-a)*(x-b)=c
(x-b)*(x-c)=a
(x-c)*(x-a)=b

We can pick two and simplify them:
(x-b)*(x-c)=a => (x-b)=a/(x-c)
(x-a)*(x-b)=c => (x-b)=c/(x-a)
a(x-a)=(x-c)c => (a-c)x=(a-c)(a+c)

So solutions can be at a-c=0 and x=a+c. x=a+c leads to a solution where two of the people are 1 year old, and the other was born in year 1. Unless we allow for a 2000 year old person, that most likely won't be the best answer. The other solution is a=c; in other words, person 1 is the same age as person 2. By similar reasoning it can be shown person 3 must also be the same age.

Now we have an easier problem: (x-a)*(x-a)=a => x^2-2a+(a^2-a)=0
The roots of this are x = a +/- sqrt(a). Or in other words, the year of the party is the birth year plus the square root of the birth year. But the year of the party is also equal to the birth year plus the age of the person. So the birth year must be a square number, and the age of the people must be the square root of that year.

So what is the numeric solution? The closest squares around 2003 are: 43^2 = 1849, 44^2 = 1936, 45^2 = 2025, so those are the birth years that will work. The party years would be 1849+43=1892, 1936+44=1980, and 2025+45=2070. So as the others said, 1980 is the most recent party year.

That being said, if the three people were being absurdly precise, and said they were 44 1/4 years old, the party could have been sometime around April 2002. But while 4 year olds might say their ages like that, I don't know many 44 years olds that would.

Posted by Ender on 2003-07-22 07:27:07