Två svåra uppgifter, kan någon?
Här kommer två mycket svåra uppgifter som det vore kul att se om någon kunde lösa Om ni klarar båda har ni troligen en IQ på 145 eller mer.
1. Suppose five dots are arranged in a three-dimensional space so that no more than three at a time can have a flat surface pass through them. If each set of three dots has a flat surface pass through them and extend an infinite distance in every direction, what is the maximum number of distinct straight lines at which these planes can intersect one another?
2. Suppose there are ants at each vertex of a triangle and they all simultaneously crawl along a side of the triangle to the next vertex. The probability that no two ants will encounter one another is 2/8, since the only two cases in which no encounter occurs is when all the ants go left, i.e., clockwise -- LLL -- or all go right, i.e., counterclockwise -- RRR. In the six other cases -- RRL, RLR, RLL, LLR, LRL, and LRR -- an encounter occurs. Now suppose that, analogously, there is an ant at each vertex of a cube and that the ants all simultaneously move along one edge of the cube to the next vertex, each ant choosing its path randomly. What is the probability that no two ants will encounter one another, either en route or at the next vertex?