Question 14
Let $p$ be a prime greater than $2$. Patricia wants to assign seven not necessarily distinct numbers from $\{1,2,\ldots,p\}$ to the black dots in the following figure, so that the product of the three numbers on any line or circle always has the same remainder upon division by $p$.
(a) Suppose Patricia uses the number $p$ at least once. What is the minimum total number of times she must use $p$?
(b) Suppose Patricia does not use $p$. In how many ways can she assign the numbers? Two assignments are different if at least one black dot receives different numbers. The figure is not rotated or reflected.
(a) Suppose Patricia uses the number $p$ at least once. What is the minimum total number of times she must use $p$?
(b) Suppose Patricia does not use $p$. In how many ways can she assign the numbers? Two assignments are different if at least one black dot receives different numbers. The figure is not rotated or reflected.