vyfukas wrote: In each field of a table there is a real number. We call such $n \times n$ table silly if each entry equals the product of all the numbers in the neighbouring fields. a) Find all $2 \times 2$ silly tables. b) Find all $3 \times 3$ silly tables. a) let the table be: $\begin{matrix} a & ab\\ ab & b \end{matrix}$ So $b=a^2 b^2$ and $a=a^2 b^2$ so $a=b$ So $b=b^4 \iff 0=b(b^3-1)$ $\begin{matrix} 0 & 0\\ 0 & 0 \end{matrix}$ $\begin{matrix} 1 & 1\\ 1 & 1 \end{matrix}$

b) let the table be: $\begin{matrix} ab & b & bc \\ a & abcd & c \\ ad & d & cd \end{matrix}$ with: $a=abcdabda$ , $b=abcdabcb$ , $c=abcdbcdc$ , $d=abcdacdd$ Trivial $\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}$ Then $1=abcd \times abd$ $1=abcd \times abc$ $1=abcd \times bcd$ $1=abcd \times acd$ So $a=b=c=d=1$