Jack and Jill each have marble collections. The number in Jack's collection in a square number.

Jack says to Jill, "If you give me all your marbles I'll still have a square number." Jill replies, "Or, if you gave me the number in my collection you would still be left left with an even square."

What is the fewest number of marbles Jack could have?

Jack's marbles , x, is a square a^2 (for some a)
Jacks + Jill's marbles , (x+y), is a square b2 (for some b)
Jack's less Jill's marbles , (x-y), is a square c^2 (for some c)
So Jack's marbles is the average of the two extremes
b^2 + c^2 = 2.a^2.
We require integers a,b,c that satisfy this equation (although in this specific problem there is the concensus that c^2 needs to be even)
...................................................
Now, consider a Pythagorean triple (ascending)
(x, y, z)..........such that x^2 + y^2 = z^2
since x, y, z all integers then (y+x) is an integer and (y-x) is an integer
let (y+x) = a
and (y-x) = b
Multiply Pytha' triple through by 2
2.x^2 + 2.y^2 = 2.z^2
Recognize the LHS can be written (y+x)^2 + (y-x)^2
(y+x)^2 + (y-x)^2 = 2.z^2
substitute a,b
a^2 + b^2 = 2.z^2 (a,b,z all integers)
So to generate a Ravi Triple (from a pythagorean);
Pythagorean triple (x y z)
Ravi triple ((y-x) (y+x) z)
So, for the smallest non trivial, unique Pyth' triple,
(3,4,5)
we get the corresponding unique Ravi triple
(1,7,5)
and Jack originally has 5^2 (25) marbles.
Obviously multiples of ravi triples also work so if the smallest square needs to be even we get
(2,14,10)
and Jack originally has 10^2 (100) marbles.
I'm not going to show how to generate Pythogorean triples. Suffice it to say they can be generated without trial and error and generators are all over the web like a rash. I just thought the link between the two types of equations was of interest.

Posted by Lee on 2003-11-04 00:53:02