Combinatorial Agriculture, part 2

Question

Your 64 acre square farm has been doing well and you decide to expand with an orchard. The orchard is also in the form of an 8-by-8 square of subplots, each of which can hold one tree. You have 3 each of 21 different types of saplings to plant, and you will build a shed on the 64th subplot.

For aesthetic reasons, for each type of tree you want one of the saplings to be exactly halfway between the other two of that type (they have to be in a straight line). You realize that if you plant each type of tree in a 1x3 rectangle of adjacent subplots, you can follow the same strategy as you did when you were planning your farm, giving you four places to build your shed.

However, the trees of each type don't have to be planted adjacent to each other. How many more spots does this allow you to build the shed in?

Answer 1

Set up Cartesian coordinates so that two diagonally opposite plots are (0, 0) and (7, 7). In each group, the sum of the 3 x coordinates of the 3 saplings in that group is a multiple of 3. Therefore this is true of all 21 groups collectively. But the total sum of the x coordinates of all 64 squares is 8*(0+...+7). Modulo 3, this is equal to 2*(0+...+7) = 2*28 = 2*1 = 2. So the shed's x-coordinate must be 2 or 5. Likewise the y-coordinate. So leoll2's answer to the early puzzle also holds for this one.