Question

Solve pls! Lots of points! Question down below!

Answer 1

Answer:1024 grid squares

Explanation:

Let's call the side length of the original piece of graph paper "x".

Since, side length of the cut out square is then (x-2) because the cut out square has to have an integer number of grid squares, and each side of the square would take up 2 grid squares on the graph paper.

The number of grid squares in the cut out square is then (x-2)^2.

The number of grid squares left on the graph paper is 124, so we have:

x^2 - (x-2)^2 = 124

Expanding the right side:

x^2 - x^2 + 4x - 4 = 124

Simplifying:

4x = 128

therefore:

x = 32

therefore original graph paper contains 32 grids i.e 32^2 = 1024

Answer 2

any of {128, 133, 140, 160, 224, 320, 380, 700, 1024}

1024 if the original grid was square

Explanation:

You want to know how many grid squares are on a piece of graph paper if a square number of grid squares can be cut from it to leave 124 grid squares.

Square graph

If the graph is square to begin with, a square of x² can be removed from its size (x+a)² to leave 124 squares. That is ...

(x +a)² -x² = 124

(a)(2x+a) = 124

The factor pairs of 124 are ...

124 = 1·124 = 2·62 = 4·31

Of these, only 2 and 62 differ by an even number, so we have a=2 and x=30.

The graph paper could be 32 units square originally, having 1024 grid squares.

Rectangular graph

There is nothing in the problem statement that requires the original graph paper be square. If it is not necessarily square, there are 13 possible shapes it could be. Those have 9 different possibilities for numbers of grid squares.

The graph paper could have originally had this number of grid squares:

• 128 = 4×32 = 8×16. The square removed is 2².
• 133 = 7×19. The square removed is 3².
• 140 = 5×28 = 7×20 = 10×14. The square removed is 4².
• 160 = 8×20 = 10×16. The square removed is 6².
• 224 = 14×16. The square removed is 10².
• 320 = 16×20. The square removed is 14².
• 380 = 19×20. The square removed is 16².
• 700 = 25×28. The square removed is 24².
• 1024 = 32×32. The square removed is 30². . . . . discussed above

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Additional comment

You may notice that removing a 24×24 square from a graph that is 25×28 leaves no border on one side.

These values were found using a search script.

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