Using two - $ 4 \times 4$ squares, three - $3 \times 3 $ squares, four - $2 \times 2$ squares and four - $1 \times 1$ squares draw a diagram to show how you can make a square using some or all of these squares together without gaps or overlaps to make a square that is as large as possible.
Explain why you cannot make a square larger than this square.
problem-solving
I can get $6 \times 6$ as shown in the first figure below. I can get close to $8 \times 8$ but would need six $1 \times 1$ squares to finish the second figure, not four. I haven't proven that $7 \times 7$ and $8 \times 8$ are impossible. The problem is that the $4 \times 4$ and $3 \times 3$ squares don't provide enough flexibility. To really prove that you would have to go through all the possible locations for the big squares and show they don't work.
