Problem 5

Let

n

be a positive integer. A pair of

n

-tuples

(a_1,\ldots,a_n)

and

(b_1,\ldots,b_n)

with integer entries is called an exquisite pair if

|a_1b_1+\cdots+a_nb_n|\le1

. Determine the maximum number of distinct

n

-tuples with integer entries such that any two of them form an exquisite pair.

Step 1 of 5: Construct

n^2+n+1

pairwise exquisite tuples

Previous step Next step

\{0\}\cup\{\pm e_i\}_{i=1}^n\cup\{e_i+e_j,\ e_i-e_j\}_{i<j} \implies 1+2n+n(n-1)=n^2+n+1

Detailed analysis

Take the zero tuple; the $2n$ tuples with a single entry equal to $1$ or $-1$ at one position; and, for each of the $\binom n2$ pairs of positions $i<j$ , the two tuples with entries $(1,1)$ and $(1,-1)$ at positions $i,j$ and zeros elsewhere. The total count is $1+2n+2\binom n2=1+2n+n(n-1)=n^2+n+1$ .