If Only I Could See In 81 Dimensions, Then This Would Be Easy

(This is a sketch, with enough information for you to fill in the details.)

First, fix an integer $k$ . Assume that all variables $a_i$ are fixed for $i \neq k$ . We want to find the value of $a_k$ , subject to $a_{k-1} \leq a_k \leq a_{k+1}$ which would maximize the expression. It is clear that the expression is a quadratic in $a_k$ , with positive leading coefficient. Hence, the maximum occurs at the endpoints, which means that $a_{k} = a_{k-1}$ or $a_k = a_{k+1}$ .

Since this is true for all $k$ , this implies that we must have $0 = a_1 = a_2 = \ldots = a_n$ , $a_{n+1} = a_{n+2} = \ldots = a_{81} = 1$ for some integer $n$ from 0 to 81. The expression thus simplifies to

$9 (81-n) + [ \text{ stuff } ],$

which is a quadratic in $n$ (with negative leading coefficient). We can easily see that the maximum happens when $n = 39$ (yes, values were specially chosen to make the maximum occur at an integer), and this would give us the value 8532.

First assumption: Must be a bunch of 0s then a bunch or 1s.

Then let x be the number of 1s in the sequence. The sum is thus:

$9x$ - Add all the ${ 1 }^{ 2 }$

$4x(81-x)$ - Add all the ${(1-0+1)}^{2}$

$.5x(x-1)$ - Add all the ${(1-1+1)}^{2}$

$.5(81-x)(81-x-1)$ - Add all the ${(0-0+1)}^{2}$

So the final equation is:

$9x+4x(81-x)+.5x(x-1)+.5(81-x)(81-x-1)$

Simple quadratic that can be simplified and the maximum found at the vertex.

After squaring we get (1+2+3+...+80)+ 89((a1)²+(a2)²+...+(a81)²)+ 4×(40(a81-a1)+39(a80-a2)+ ...+2(a43-a39)+(a42-a40)) -2(a81(a1+a2+...+a80)+ a(80(a1+a2+...+a79)+ +...+a3(a1+a2)+a2×a1

So first we can conclude a81=a80=…=a42=1 a40=…=a2=a1=0 For a41 we try 0 or 1 and we see that sum is greater when a41=1

Now,we can calculate value as 80×81/2+89×41+4×(40+39+...+1) -2(40+39+...+1)=40×81+40×89+89+ +40×41=40×211+89=8529 But if we put x40=1 instead of 0 We'll get 40×81+42×89+4×(40+...+1) -4-2(41+...+1)= 40×81+40×89+178+41×40-4-82= 40×211+92=8440+92=8532 If we put a39=1 instead of zero we'll get 8352+89-2×42-4×2=8529 So max is 8352 when a1=a2=…=a39=0 a40=a41=…=a81=1