Given an n x n grid, where n is always a power of two. Prove that for such a grid you can always cover all tiles but one space with L shaped tiles.

Divide an equilateral triangle into a number of triangles that are similar to the original triangle (ie similar in shape but possibly different size and possibly rotated) so that they lie in k rows in the original triangle. If you select k = 2, 4, 8 you get the following figures (for eg k = 6 you remove the two bottom rows in the right figure, for k = 3 you delete the bottom row in the middle figure, etc.). Let ak be the number of triangles when you have k rows and consider the case k = 2n, n ≥ 0. A. Determine the values for a2m where n = 1, 2, 3, and 4. B. Using the results in task A, determine a recurrence relation for a2n, where n ≥ 0.

Let V = (a, b, c, d), such that a = 2b − c, b = 2d − 3c, and d = a − b + c. Find the dimension of V .

This is a new version of the question. Make sure you start new workings. When this net is folded into a cube, which edge joins with edge X? A X F B E C D