Stacking rectangles

If there are $N$ 1 by 51 rectangles, then the area that they will take up is $51 \cdot N$ . Since $100 \times 100 \div 51 < 196.1$ , we can cut out at most 196 such 1 by 51 rectangles. It remains to show that this can be done.

[Image reproduced in the question above.]

Refer to the image above. Take 49 vertical 1 by 51 rectangles stacked horizontally left to right. Place 49 horizontal rectangles stacked vertically over each other to the right. Place 49 horizontal rectangles stacked vertically above the initial vertical rectangle. Place 49 vertical rectangles in the top right corner. This leaves a 2 by 2 empty space in the middle, and is a packing of $49 \times 4 = 196$ 1 by 51 rectangles.