Can This Mathematical Permutation Problem Be Generalized for Any 4n Terms?

[dapet]

Determine the permutation (x_1,x_2,...,x_40) of numbers 1,2,...,40 such that the expression x_1 + x_2 / x_3 - x_4 * x_5 + x_6 / x_7 - x_8 * x_9 + ... + x_38 / x_39 - x_40
has the maximal possible value. (the operators +,/,-,* alternate regularly.) It can be generalized for 4n (where n is an arbitrary natural)?

But I must confess, that I have a lot of problems with the simpler task... I think that more than one permutation give the same maximum. I have one suggestion, but I can't prove that it belongs to the best of all.
It's based on:
let (a,b,c,d,e,f) be a permution of (1,2,3,4,5,6)
minimum of ab + cd + ef occurs for 6*1 + 2*5 + 3*4
maximum of a/b + c/d + e/f occrus for 6/1 + 5/2 + 4/3
But the generalization...

Thanks.

 

[dapet]

I think that the maximum looks like:
40+(39/20)+(38/21)+...(30/29)-(1*18)-(2*17)-...-(10*9)-19 and the generalization is (according to my opinion) similar... It's easy to prove that numbers 40,39,...,30 must be in numerators of fractions for reach the maximum, but I don't know how to prove that the "position" of other numbers as optimal as possible for reach the maximum...

Thanks for each help that I really need.

 

[M98Ranger]

Your approach seems to be on the right track. In fact, you have found a permutation that gives the maximum value for 6 terms, and this can be extended to any number of terms. For the generalization, we can use the same logic as you did for 6 terms, but we need to consider the pattern for 4n terms.

Let (a,b,c,d) be a permutation of (1,2,3,4)
Minimum of ab + cd occurs for 4*1 + 2*3
Maximum of a/b + c/d occurs for 4/1 + 3/2

So, for 8 terms, we have:
Minimum of ab + cd + ef + gh occurs for 4*(1+5) + 2*(3+7)
Maximum of a/b + c/d + e/f + g/h occurs for 4/(1+5) + 3/(3+7)

Similarly, for 12 terms, we have:
Minimum of ab + cd + ef + gh + ij + kl occurs for 4*(1+5+9) + 2*(3+7+11)
Maximum of a/b + c/d + e/f + g/h + i/j + k/l occurs for 4/(1+5+9) + 3/(3+7+11)

This pattern can be generalized for any number of terms, where the minimum will occur for the sum of the first n/2 terms and the maximum will occur for the sum of the first n/2 terms divided by the sum of the last n/2 terms.

So, for 40 terms, we have:
Minimum of ab + cd + ... + yz occurs for 4*(1+5+...+37) + 2*(3+7+...+39)
Maximum of a/b + c/d + ... + y/z occurs for 4/(1+5+...+37) + 3/(3+7+...+39)

This gives us a permutation of (1,2,...,40) that will give the maximum value for the given expression. However, as you mentioned, there may be multiple permutations that give the same maximum value. This can be proven using mathematical induction.

Therefore, the generalization for 4n terms holds true and your approach is correct.