第4题

译文：An

n\times n

矩阵 whose entries belong 到

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

是called a silver 矩阵 if, 对于每个

i

i

th 行 together 满足

i

th 列 contains 所有elements 的

S

. 证明 (a) there 是no silver 矩阵对于

n=1997

; (b) silver matrices exist 对于infinitely many

n

第 4/4 步：Verify 且iterate

上一步下一步

通俗地说

Verify 且iterate

1,2,\ldots,4n-1

详细分析

F或每个matched 行且列的 $B$ , unshifted blocks supply $1,\ldots,2n-1$ 且shifted blocks supply $2n,\ldots,4n-1$ exactly once (special diagonal entry supplies missing $2n$ ). 故 $B$ 是silver. The $1\times1$ 矩阵 $[1]$ 是silver, so repeated doubling gives silver matrices 的sizes $1,2,4,8,\ldots$ 译文：, infinitely many.