My quick bit of algebra also indicates that you can't do it with integral values:
x * 1 + y * 4 + z * 9 -> 25 x + y + z -> 9
z <= 1 because you can't fit two 3x3 squares in without an overlap. And I'm not bothering with any 4x4 squares, since simple inspection means the minimum total number of squares is then 10
if z = 1, then
x + 4y -> 16 x + y -> 8
And therefore 3y -> 8 - not an integral value.
Contrarily, if z = 0
x + 4y -> 25 x + y -> 9
And 3y -> 16, still not an integral solution.
So my solution is to abandon integer values, since the problem does not specify them, and give the same design as the original one shown (just each 'unit' is 5/6 as large). You can rotate the bottom right quadrant or exchange it with one of the others as desired.
no subject
My quick bit of algebra also indicates that you can't do it with integral values:
x * 1 + y * 4 + z * 9 -> 25
x + y + z -> 9
z <= 1 because you can't fit two 3x3 squares in without an overlap. And I'm not bothering with any 4x4 squares, since simple inspection means the minimum total number of squares is then 10
if z = 1, then
x + 4y -> 16
x + y -> 8
And therefore 3y -> 8 - not an integral value.
Contrarily, if z = 0
x + 4y -> 25
x + y -> 9
And 3y -> 16, still not an integral solution.
So my solution is to abandon integer values, since the problem does not specify them, and give the same design as the original one shown (just each 'unit' is 5/6 as large). You can rotate the bottom right quadrant or exchange it with one of the others as desired.