Saved Bookmarks
| 1. |
. There are infinite black and white dots on a plane. Prove that the distance between one black dot and one white dot is one unit. |
|
Answer» (Alternately, you could prove the new STATEMENT by contradiction. Pick any BLACK point. All points in distance 1 from that point have to be black. This is the circle with radius 1. All points in distance 1 from those points have to be black as well. Here we can observe that the set of all points known to be black at this moment is the entire disc of radius 2 centered where we started. Continuing this ARGUMENT, we can now grow the black disc indefinitely and thus prove that the entire plane has to be black, which is the contradiction we seek. Of COURSE, this is basically the same proof as above, just seen from a different point of view.) |
|