Brain Buster #3 - Mensa Brain Twisters

 Mensa brain twisters.  

These have given some of the best brains in Mensa fits. There are no time limits and one or two may require computer programming skills. Readers can work out parts of each subset maybe a few each day, and see how they do.  If one at first seems too hard, come back to it.

1)

Arrange the 6 L-shaped pieces above into a 4 x 6 grid such that:

- Each row has the same number of stars

-  Each column has the same number of stars.

- No diagonal has 4 stars.

The pieces may be rotated but not reflected.

2) Prove all square numbers (1,4,9,16, 25...) are either divisible by three OR have a remainder of one when divided by three.

3) Find all solutions for positive integers a and b (a > 1):

(a + 2)! a! (a - 2)! = b!

Then prove there are no other solutions.

4) Given a rectangle of length  (Ö2) x  and height x, we have one circle and two semi-circles inscribed within it,  and all three intersect at two points, as shown below:

Find the area of either green crescent shape between the two arcs in terms of x. 

5) Replace the letters A-L in the equation below with the integers 0-11 so that the math is correct.

A^B + C^D = E ^F + G^H + I^J + K^L