Correction: The statement beginning with ***** is obviously in error. (What was I thinking?) The correct statement now follows the wrong one. ---------------------------------------------------------------------------- ------------------------------ (I don't know whether either of these problems has been solved. Here are the 3D versions of what could asked of any dimension.) 1. Analogous to a recent question posed here about triangles: Let A_n denote a regular tetrahedral arrangement of points in 3-space, n to an edge. (The layers successively contain 1, 3, 6,... points.) Define a "tetrad" to be any four mutually adjacent points of A_n. QUESTION: For which n is it possible to partition A_n into disjoint tetrads ??? Call such an n "tetrizable". *****(WRONG:) Clearly, any such n = 0 (mod 4). Clearly, any such n implies that the size of A_n is divisible by 4. It's not hard to show that #(A_n) is divisible by 4 exactly when n = 0,2,4,6, or 7 (mod 8). QUESTION: More concretely, n = 2 is the first tetrizable number. What is the next one (if any) ??? --------------------------------------------------------------- 2. Given any arrangement of congruent spheres in 3-space where any two are either disjoint or tangent, consider the graph formed by letting each sphere define a node, and each tangency an edge. Call such a graph a "packing graph". QUESTION: What is the maximum chromatic number of a packing graph (if the maximum exists) ??? ------------------------------------------------------------- --Dan Asimov