### Brain Teasers

# Marbles in Squares

Lester and Terance each have marble collections. The number in Lester's collection is a square number (1,4,9,16, etc). Lester says to Terance, "If you give me all of your marbles I'll still have a square number." Terance replies, "If you gave me the number in my collection you would still be left with a square." What is the least number of marbles Lester has?

### Answer

Lester has 25 marbles and Terance has 24 marbles.Hide Answer Show Answer

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## Comments

Good teaser.

Or, Lester has 1 marble, and Terrace has a marble collection of marble slabs. Therefore, he can't give him any "marbles", so lester always has that 1 marble which is a square. Though, that would be quite a bit of a stretch.

There is a way to calculate every possible solution to this teaser.

For every positive integer N, there exists a solution for each factor F of N.

Where

X = (2F*(F-1)+1)*(N/F)

the number of marbles that Lester has is:

L = X^2

The number of marbles Terance has is:

T = L - (X-2N)^2

The minimum solution is for N = 1 and F = 1:

X = (2*1*(1-1)+1)*(1/1) = 1

L = 1^2 = 1

T = 1 - (1-(2*1))^2 = 0

All solutions with F = 1 will have zero marbles for Terance. The minimum solution where F is greater than 1 is N = 2, F = 2:

X = (2*2*(2-1)+1)*(2/2) = 5 * 1 = 5

L = 5^2 = 25

T = 25 - (5-(2*2))^2 = 25 - 1 = 24

The solution for N = 6, F = 3 is:

X = (2*3*(3-1)+1)*(6/3) = 13 * 2 = 26

L = 26^2 = 676

T = 676 - (26-(2*6))^2 = 676 - 14^2 = 676 - 196 = 480

(676 - 480) = 196 = 14^2

(676 + 480) = 1156 = 34^2

For every positive integer N, there exists a solution for each factor F of N.

Where

X = (2F*(F-1)+1)*(N/F)

the number of marbles that Lester has is:

L = X^2

The number of marbles Terance has is:

T = L - (X-2N)^2

The minimum solution is for N = 1 and F = 1:

X = (2*1*(1-1)+1)*(1/1) = 1

L = 1^2 = 1

T = 1 - (1-(2*1))^2 = 0

All solutions with F = 1 will have zero marbles for Terance. The minimum solution where F is greater than 1 is N = 2, F = 2:

X = (2*2*(2-1)+1)*(2/2) = 5 * 1 = 5

L = 5^2 = 25

T = 25 - (5-(2*2))^2 = 25 - 1 = 24

The solution for N = 6, F = 3 is:

X = (2*3*(3-1)+1)*(6/3) = 13 * 2 = 26

L = 26^2 = 676

T = 676 - (26-(2*6))^2 = 676 - 14^2 = 676 - 196 = 480

(676 - 480) = 196 = 14^2

(676 + 480) = 1156 = 34^2

Good teaser! Even more amazing general solution Java. (General solutions rock). I started to look for a pattern trying A square number for Lester and for what Lester had remaining after the gift. (I tried (3, 1) (4, 1 and (5, 1) which gave me the solution of 24 for Terance. Since 24 + 25 = 49 a square number that was it and I really needed to look no further.

But the general solution blew me away.

But the general solution blew me away.

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