Problem 4

n\times n

matrix whose entries belong to

S=\{1,2,\ldots,2n-1\}

is called a silver matrix if, for each

i

, the

i

th row together with the

i

th column contains all elements of

S

. Show that (a) there is no silver matrix for

n=1997

; (b) silver matrices exist for infinitely many

n

Step 2 of 4: Rule out odd

n

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In plain words

Rule out odd $n$

n(2n-1)+d

Detailed analysis

On the other hand, the $n$ row-column pairs each contain one copy of all $S$ , plus one extra copy for every diagonal entry. If $n$ is odd, every element initially occurs an odd number of times; at most $n$ diagonal entries can change parity, so at least one of the $2n-1>n$ elements remains odd. Thus no silver matrix exists for odd $n$ , in particular $n=1997$ .