{Baker, Cooper, Fletcher, Miller, Smith};
(Unequal @@ %) && (And @@ (0 < # < 6 & /@ %)) &&
Baker < 5 &&
Cooper > 1 &&
1 < Fletcher < 5 &&
Miller > Cooper &&
Abs[Smith - Fletcher] > 1 &&
Abs[Cooper - Fletcher] > 1 //
Reduce[#, %, Integers] &