The diagram shows how it is possible to place 7 X’s and 18 O’s in a 5 x 5 array so that no two of the X’s are on the same diagonal line.

O	X	O	O	O
O	O	O	X	X
X	O	O	O	O
X	O	O	X	O
O	O	O	X	O

 a        Is it possible to place 8 X’s and 17 O’s in a 5 x 5 array so that no two of the X’s are on the same diagonal line? Justify your answer.

b       Give, with proof, the maximum number of X’s it is possible to place in a 5 x 5 array (with O’s filling all the other places in the array) so that no two of the X’s are on the same diagonal line?

In the diagram above, every O in the array is on the same diagonal line as at least one of the X’s.

c        Is it possible to arrange fewer than 7 X’s in a 5 x 5 array (with O’s filling all the other places in the array) so that every O in the array is on the same diagonal line as at least one of the X’s? Justify your answer.

 d       Give, with proof, the minimum number of X’s it is possible to place in a 5 x 5 array  (with O’s filling all the other places in the array) so that every O in array is on the same diagonal line as at least one of the X’s?