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?
Let J= Jill's marbles
Let a^2=Jack's marbles
From the problem we have:
a^2+J=b^2, a^2-J=(2c)^2, since c is required to be even;
Adding: 2a^2=b^2+(2c)^2
But b>a, so, say:
2a^2=(a+n)^2+(2c)^2
The smallest value c can take is 1, so let:
a^2=2an+n^2+4. The smallest value {a,n} is {2,0}
This is a Pellian with 10 as the next solution for a:
2(10)^2=(10+n)^2+(2*1)^2, gives n=4, so b=14.
Checking:
100+J=196, J=96
100-J=4, J=96
Jack had 100 marbles, Jill had 96.
Edited on May 20, 2022, 12:33 am
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Posted by broll
on 2022-05-20 00:17:08 |
Possible Solution
