Let Y denote the union of 3 unit intervals sharing one common endpoint. Consider the *cartesian square* Y x Y. Since Y is the union of 3 intervals I_j, j=1,2,3, Y^2 must be the union of 9 squares Q_jk = I_j x I_k that may intersect each other, but only along a common edge or vertex.) Let G be the graph consisting of all the edges of Y x Y. PUZZLE: ------- How many distinct simple closed curves occur in G ??? Extra credit section: ------------------------------------------------------------------- We can also look at the space that is the nth cartesian power of Y: Y^n = Y x Y x ... x Y (n times), and ask: ----- What is the "embedding dimension" E(n) of Y^n, defined as the least d in Z+ such that Y^n is homeomorphic to a subset of Euclidean space R^d ??? ---- Since Y is (homeomorphic to) a subset of R^2, clearly Y^n is a subset of R^(2n). Question: ----- For instance, can Y^2 be embedded in R^3 ??? ----- ***> A proof or a counterexample is sought. ------------------------------------------------------------------- —Dan