Question

You have two rectangular grids of 1cm3 cubes, one measuring n + 1 by n + 6cm, and the other measuring n + 3 by n + 3cm, where n is a positive integer. Find and simplify an expression for the total number of 1cm^3 cubes in the two rectangular grids combined. Show that if you take apart the rectangles and recombine the 1cm3 cubes you will always be able to form a rectangle with no cubes left over (where the rectangle will not simply be a straight line of cubes.

Answer 1

Explanation:

The total number of cubes in the first rectangular grid is (n+1) x (n+6) x 1 = (n^2+7n+6) cubes.

The total number of cubes in the second rectangular grid is (n+3) x (n+3) x 1 = (n^2+6n+9) cubes.

Therefore, the total number of cubes in both rectangular grids combined is (n^2+7n+6) + (n^2+6n+9) = 2n^2 + 13n + 15.

To show that you can always form a rectangle with no cubes left over, we can note that the dimensions of the first grid are (n+1) x (n+6) and the dimensions of the second grid are (n+3) x (n+3).

If we take apart the first grid and recombine the cubes, we can form a rectangle with dimensions (n+1) x 6.

If we take apart the second grid and recombine the cubes, we can form a square with dimensions (n+3) x (n+3).

Now, we can combine the two rectangles by placing the square on top of the rectangle with dimensions (n+1) x 6, and aligning their edges. This gives us a new rectangle with dimensions (n+3) x (n+7).

We can see that the total number of cubes used to form this new rectangle is equal to the sum of the cubes used to form the original two rectangles, and no cubes are left over. Therefore, we have shown that we can always form a rectangle with no cubes left over by taking apart and recombining the cubes from the two original rectangles.