Oh, I was bored, so here's another one.
I secretly pick four points. I then give you six distinct points that are the midpoints of every pair of mine. How many possible configurations could I have started with? Construct them.
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×There are two possible starting configurations.
Suppose I had given you the three midpoints of pairs of three points. Then you could easily reconstruct the original three points; they form a triangle twice as big with sides parallel to the triangle I gave you. Therefore, in the case of the stated problem, if you know three points that are the midpoints in a triangle with corners at three of my points, then you could find three of my points.
Now notice that the configuration of six points I give you must be centrally symmetric about the center of gravity of my four points. You can then pair off these points so that the points in a pair lie halfway between two disjoint pairs of my points. Take three points that each lie in a different such pair, and assume they form the midpoints of the sides of a triangle of my points, and find three points that you think are mine as in the previous paragraph. They form a triangle that is twice as big and with sides parallel to the triangle with corners at the three unused points by the central symmetry, so finding the center of the dilation that takes one to the other gives the fourth point and a configuration. In this procedure, there are eight different triangles you could have taken to start, but each configuration has four triangles in it, so you get two configurations. The fact that they're different follows from the fact that the six points I gave you were distinct.
), then it's okay. But before I go to bed, I'm locking this topic. The reason is that people need to be able to vote on entries that aren't changing.