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 1 of 4: Count entries two ways

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In plain words

Count entries two ways

2n-1

Detailed analysis

List all entries row by row and then column by column. Every matrix entry is listed twice, so each element of $S$ occurs an even number of times in this combined list.