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Fair and Square (Posted on 2003-11-03) Difficulty: 3 of 5
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?

See The Solution Submitted by Ravi Raja    
Rating: 2.3333 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Possible Solution | Comment 24 of 25 |
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
  Posted by broll on 2022-05-20 00:17:08

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