Interestingly, it seems that nobody's studied the WORST rational approximations to pi! I'd better explain what I mean by this before anyone shouts "Hey, a *million* is a really bad rational approximation to pi!" Let nint(x) denote the nearest integer to x. For each denominator q, look at the discrepancy |q Pi - p| where p = nint(Pi q); call this disc(q). The best rational approximations to pi are those for which disc(q) is smaller than disc(1),...,disc(q-1), so it's natural to define the worst rational approximations to pi as those for which disc(q) is greater than disc(1),...,disc(q-1). If I haven't goofed, the denominators of the worst approximations to pi are *1,2,3,4,11,18,25,32,...* (a sequence that isn't in OEIS). Before I compute more terms, can anyone corroborate my numbers? It seems improbable to me that nobody's thought to look at this before. (If we play this same game with the golden ratio, we get *1,4,17,72,305,1292,...* which is in the OEIS (A001076 <http://oeis.org/A001076>).) Jim