Question 37
Fred the Four-Dimensional Fluffy Sheep is walking in 4-dimensional space. He starts at the origin. Each minute, he walks from his current position $(a_{1}, a_{2}, a_{3}, a_{4})$ to some position $(x_{1}, x_{2}, x_{3}, x_{4})$ with integer coordinates satisfying
$$(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}+(x_{3}-a_{3})^{2}+(x_{4}-a_{4})^{2}=4$$
and
$$\left| (x_{1}+x_{2}+x_{3}+x_{4})-(a_{1}+a_{2}+a_{3}+a_{4}) \right|=2.$$
In how many ways can Fred reach $(10,10,10,10)$ after exactly 40 minutes, if he is allowed to pass through this point during his walk?
$$(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}+(x_{3}-a_{3})^{2}+(x_{4}-a_{4})^{2}=4$$
and
$$\left| (x_{1}+x_{2}+x_{3}+x_{4})-(a_{1}+a_{2}+a_{3}+a_{4}) \right|=2.$$
In how many ways can Fred reach $(10,10,10,10)$ after exactly 40 minutes, if he is allowed to pass through this point during his walk?