Question 64
How many 6-tuples $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}\right)$ are there such that each of $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ is from the set $\{1,2,3,4\}$ and the six expressions
$$
a_{j}^{2}-a_{j} a_{j+1}+a_{j+1}^{2}
$$
for $j=1,2,3,4,5,6$ (where $a_{7}$ is to be taken as $a_{1}$) are all equal to one another?
$$
a_{j}^{2}-a_{j} a_{j+1}+a_{j+1}^{2}
$$
for $j=1,2,3,4,5,6$ (where $a_{7}$ is to be taken as $a_{1}$) are all equal to one another?