Several exchange constants J(i) between Mn2+ ions which are not neares
t neighbors were determined in Zn1-xMnxX (X = S, Se, Te) from magnetiz
ation steps at 20 mK. When the J(i)'s are listed in order of decreasin
g size, ratios between successive J(i)'s are material dependent, and d
iffer from all predictions. The measured J(i)'s were identified by com
paring the magnetization curves with simulations which assumed a rando
m Mn distribution. Contrary to existing theories the second-largest ex
change constant is not J(2) between next-nearest neighbors. The most l
ikely alternative is J(4), between fourth neighbors.