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 4 of 4: Verify and iterate

Previous step Next step

In plain words

Verify and iterate

1,2,\ldots,4n-1

Detailed analysis

For each matched row and column of $B$ , the unshifted blocks supply $1,\ldots,2n-1$ and the shifted blocks supply $2n,\ldots,4n-1$ exactly once (the special diagonal entry supplies the missing $2n$ ). Thus $B$ is silver. The $1\times1$ matrix $[1]$ is silver, so repeated doubling gives silver matrices of sizes $1,2,4,8,\ldots$ , infinitely many.