I've noticed some nice number theoretic properties of the median which are probably well known, but I'd be interested in references A sequence of real numbers (a_1,a_2, . . ,a_n) has a PERFECT MEDIAN if there is an index m such that a_1+. . +a_m = a_m+ . . +a_n The sequence has a DOUBLE MEDIAN if there exists an index m such that a_1+ . . +a_m = a_(m+1)+ . +a_n Example: The constant sequence (c) has a perfect median if n is odd, double medians if n is even. For which n does the sequence (1,2, . ,n) have a perfect median? If n=1 then m=1 is a perfect median x If n=8 then m=6 is a perfect median x xx xxx xxxx xxxxx mmmmmm xxxxxxx xxxxxxxx The next case is n=49,m=35. For which n does the sequence (1,2, . ,n) have a double median? If n=3 then 2 and 3 are medians x xx ======== xxx If n=20 then 14 and 15 are medians x xx xxx xxxx xxxxx xxxxxx xxxxxxx xxxxxxxx xxxxxxxxx xxxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxxx xxxxxxxxxxxxxx _======================== xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx The next case is n=119 with medians 84 and 85. It turns out one can parameterize all solutions in both cases by solutions of a suitable Pell equation. Again, references please. David