Mathematica's Minimize function

Is it true that Mathematica's Minimize function does not allow constraints like Mod[x,2]==0? I am tr开发者_开发问答ying to solve a MinuteMath puzzle with Mathematica:

What is the smallest possible average of four distinct positive even integers?

My "solution" looks like this:

vars = Subscript[x, #] & /@ Range[4];
cond = Apply[And, Mod[#, 2] == 0 & /@ vars] && 
   (0 < Subscript[x, 1]) &&
   Apply[And, Table[Subscript[x, i] < Subscript[x, i + 1], {i, 1, 3}]];
Minimize[{Mean[vars], cond}, vars, Integers] 

but Minimize returns unevaluated. Additional question: Can I use EvenQ for defining the constraints? Problem is, EvenQ[x] returns False for undefined expressions x.

Clearly, this doens't require Mathematica but, in answer to your question, it seems that Minimize doesn't like the mods. You could build it into the forumula, though, like so:

Minimize[{(2 x1 + 2 x2 + 2 x3 + 2 x4)/4,
  0 < x1 < x2 < x3 < x4}, {x1, x2, x3, x4}, Integers]

A clear overkill for this problem, but useful to show some tricks.

Note that:

 Exists[x, Element[x, Integers] && n x == y]

can be used as an alternative to

  Mod[y,n] == 0

So:

Minimize[{(x1 + x2 + x3 + x4)/4, 0 < x1 < x2 < x3 < x4 && 
   Exists[x, Element[x, Integers] && 2 x == x1] &&
   Exists[x, Element[x, Integers] && 2 x == x2] &&
   Exists[x, Element[x, Integers] && 2 x == x3] &&
   Exists[x, Element[x, Integers] && 2 x == x4]
  },
 {x1, x2, x3, x4}, Integers]  

-> {5, {x1 -> 2, x2 -> 4, x3 -> 6, x4 -> 8}}  

Or perhaps more elegant:

s = Array[x, 4];  
Minimize[{  
  Total@s,  
  Less @@ ({0} \[Union] s) &&  
   And @@ (Exists[y, Element[y, Integers] && 2 y == #] & /@ s)},
s, Integers]

--> {20, {x[1] -> 2, x[2] -> 4, x[3] -> 6, x[4] -> 8}}

Minimize is not good for this question, it gives a wrong solution:

2 4 8 10

The good solution is:

2 4 6 12

"Select" accept Mod function it's better than Minimize
Minimize gives the wrong solution.

Play this programe and see:

Res=Select[Permutations[Range@20 {4}],Mod[ #[[1]],2]==0 &&Mod[ #[[2]],2]==0&&Mod[#[[3]],2]==0 &&Mod[#[[4]],2]==0&&Mod[(#[[1]]+#[[2]]+#[[3]]+#[[4]])/4,2]==0 &];Sort[Res,Less];Res[[1]]