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Omni-MATH / Fred the Four-Dimensional Fluffy Sheep is walking in 4 -dimensional space. He…

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problem

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 (a1,a2,a3,a4)\left(a_{1}, a_{2}, a_{3}, a_{4}\right) to some position (x1,x2,x3,x4)\left(x_{1}, x_{2}, x_{3}, x_{4}\right) with integer coordinates satisfying (x1−a1)2+(x2−a2)2+(x3−a3)2+(x4−a4)2=4\left(x_{1}-a_{1}\right)^{2}+\left(x_{2}-a_{2}\right)^{2}+\left(x_{3}-a_{3}\right)^{2}+\left(x_{4}-a_{4}\right)^{2}=4 and |(x1+x2+x3+x4)−(a1+a2+a3+a4)|=2\left|\left(x_{1}+x_{2}+x_{3}+x_{4}\right)-\left(a_{1}+a_{2}+a_{3}+a_{4}\right)\right|=2. In how many ways can Fred reach (10,10,10,10)(10,10,10,10) after exactly 40 minutes, if he is allowed to pass through this point during his walk?
Plain-text mathematical notation (without MathML)
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 (a₁,a₂,a₃,a₄) to some position (x₁,x₂,x₃,x₄) with integer coordinates satisfying (x₁−a₁)²+(x₂−a₂)²+(x₃−a₃)²+(x₄−a₄)²=4 and |(x₁+x₂+x₃+x₄)−(a₁+a₂+a₃+a₄)|=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?
Original LaTeX notation
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 $\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ to some position $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ with integer coordinates satisfying $\left(x_{1}-a_{1}\right)^{2}+\left(x_{2}-a_{2}\right)^{2}+\left(x_{3}-a_{3}\right)^{2}+\left(x_{4}-a_{4}\right)^{2}=4$ and $\left|\left(x_{1}+x_{2}+x_{3}+x_{4}\right)-\left(a_{1}+a_{2}+a_{3}+a_{4}\right)\right|=2$. 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?

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Source and history

task_id
8861403e-b05a-5732-955f-ddb999e2d3f3
task_key
test--8861403e-b05a-5732-955f-ddb999e2d3f3
task_revision_id
1
config / split
/ test

Official source

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