finds 5 "antimony" solutions. In[405]:= Reduce[a > b > c > d && a b c d/10^6 == a + b + c + d == 711, {a, b, c, d}, Integers] // tim During evaluation of In[405]:= 3369.569497,6 Out[405]= (a == 316 && b == 150 && c == 125 && d == 120) || (a == 600 && b == 250 && c == -60 && d == -79) || (a == 900 && b == 316 && c == -5 && d == -500) || (a == 1185 && b == 50 && c == -24 && d == -500) || (a == 1250 && b == 15 && c == -80 && d == -474) || (a == 1500 && b == 25 && c == -24 && d == -790) In[406]:= ToRules@% Out[407]= Sequence[ {a -> 316, b -> 150, c -> 125, d -> 120}, {a -> 600, b -> 250, c -> -60, d -> -79}, {a -> 900, b -> 316, c -> -5, d -> -500}, {a -> 1185, b -> 50, c -> -24, d -> -500}, {a -> 1250, b -> 15, c -> -80, d -> -474}, {a -> 1500, b -> 25, c -> -24, d -> -790}] In[410]:= {a b c d, a + b + c + d} /. List@%407 Out[410]= {{711000000, 711}, {711000000, 711}, {711000000, 711}, {711000000, 711}, {711000000, 711}, {711000000, 711}} (Dividing these "pennies solutions" by 100. will give $7.11.) Does anyone know the origin of this problem? --rwg I once found a (messy) addition formula yielding an infinitude of rational solutions, including some with nothing smaller than half cents.