1995 Putnam B3

To each positive integer with $n^{2}$ decimal digits, we

associate the determinant of the matrix obtained by writing the

digits in order across the rows. For example, for $n=2$, to the

integer 8617 we associate $\det \left(

\begin{array}{cc} 8 & 6 \\

1 & 7 \end{array} \right) = 50$. Find, as a function of$n$, the

sum of all the determinants associated with $n^{2}$-digit

integers. (Leading digits are assumed to be nonzero; for example,

for $n=2$, there are 9000 determinants.)

对于每个具有 $n^{2}$ 位小数的正整数，我们

关联通过编写获得的矩阵的行列式

各行中的数字按顺序排列。例如，对于 $n=2$，

整数 8617 我们将 $\det \left(

\begin{数组}{cc} 8 & 6 \\

1 & 7 \end{array} \right) = 50$。找到$n$ 的函数，

与 $n^{2}$ 位数字相关的所有决定因素的总和

整数。 (假定前导数字非零；例如，

对于 $n=2$，有 9000 个行列式。)