How many total squares of all sizes are present in a 4x4 chessboard grid?
Total squares in an n x n grid = $\sum_{$$k=1}^{n} k^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$.
How many total rectangles of all sizes (including squares) are present in a 3x3 grid?
Rectangles in an m x n grid = C(m+1, 2) * C(n+1, 2). For 3x3: C(4, 2) * C(4, 2) = 6 * 6 = 36.
A 3D object is rotated 90 degrees clockwise about the vertical Z-axis, then reflected across the horizontal XY-plane. Which of the following describes its final orientation relative to initial?
Rotation around vertical Z-axis is yaw (90 deg CW). Reflection across horizontal plane flips top/bottom (vertical inversion).
Opposite faces of a standard six-sided die always sum to 7. If '1' is on top and '2' is facing front, what number is on the bottom face?
Top and bottom are opposite faces. Since top is 1, bottom = 7 - 1 = 6.
In the same standard die (opposite sum = 7) with '1' on top and '2' facing front, what number is on the back face?
Front and back are opposite faces. Since front is 2, back = 7 - 2 = 5.