Some interesting ways to represent numbers in the interval (a,b): Given (say positive) reals a < T < b, with T not of the form (K*a + L*b)/M for any integers K >= 0, L > = 0, M >= 1, create a sequence S_n of a's and b's as follows: S_1 = a. Let avg_n denote (S_1 + . . . + S_n) / n If avg_n > T, then S_(n+1) = a; if avg_n < T, then S_(n+1) = b; if avg_n = T, the sequence terminates. E.g., let a = 1, b = 2, and T = sqrt(2). Then what is the sequence of 1's and 2's ??? --------------------------------------------------------- This is of course equivalent to doing the analogous thing with multiplication and the geometric mean; this view often leads to more interesting-sounding questions. E.g., let P_0 = 1. If (P_1*...*P_n) < e^n, then P_(n+1) = 3; else P_(n+1) = 2. This last example brings up some questions: Is there a simple formula for P_n ? Are the lengths of runs on 2 or on 3 unbounded? Does the ratio R_n = (# 2's in P_1 through P_n) / (# 3's in P_1 through P_n) approach a limit as n -> oo ? If not, does log(R_n) remain bounded as n -> oo ? Does {R_n} have a lim sup or lim inf, and if so what are they? Etc., etc. --Dan