cut into 5 vertically (4 cuts) and into 4 horisontally (5 cuts) 5x4= 20 identical pieces
7
an infinite number
With 10 straight cuts, you can create a maximum of 56 pieces of pie. This is based on the formula for the maximum number of pieces ( P(n) = \frac{n(n + 1)}{2} + 1 ), where ( n ) is the number of cuts. For 10 cuts, substituting into the formula gives ( P(10) = \frac{10 \times 11}{2} + 1 = 56 ).
8
We'll say 9 cuts.
9 pieces
When a cube is cut by 15 cuts, it can produce a maximum of 27 identical pieces. Each cut can create at most 2 identical pieces, so with 15 cuts, you can get 2 x 15 = 30 pieces. However, 3 of these pieces will be removed as they are the corners of the cube, leaving you with 30 - 3 = 27 identical pieces.
7
To cut a cube into 504 identical pieces, you would need to make 503 cuts. Each cut divides the cube into two pieces, so the first cut creates 2 pieces, the second cut creates 4 pieces, the third cut creates 8 pieces, and so on. Therefore, to reach 504 pieces, you would need to make 503 cuts.
If you do not re-stack the pieces, then 15 cuts.
If no cut intersects any previous cuts, then you can just slice it into 14 pieces.
324
26
In one sense you cannot. The cakes would have a different number of faces which were part of the original faces. To that extent the pieces will not be identical. If such pieces are considered identical, and if the cake pieces can be stacked before cutting, then 9 cuts will suffice. Without stacking, 12 cuts are required. If the cake can be stacked and cut, and a little wastage (less than 2.5%) is pemitted, then 7 cuts will be enough.
26
7
An infinite number of ways. Cut along a line from anywhere on a side to the centre of the square. Make three more cuts, at 90, 180 and 270 degrees to the first at the centre. Each point on a side of the square will give rise to a different set of four identical pieces of the square. And there are an infinite number of points on the side of the square. So an infinite number of answers to the question.