On the continual Rubik’s cube

Let ff and gg be two continuous functions on the unit sphere Sn−1Sn−1 in RnRn, n≥3n≥3, and let their restrictions to any one-dimensional great circle EE coincide after some rotation ϕEϕE of this circle: f(ϕE(θ))=g(θ)∀θ∈Ef(ϕE(θ))=g(θ)∀θ∈E. We prove that in this case f(θ)=g(θ)f(θ)=g(θ) or f(θ)=g(−θ)f(θ)=g(−θ) for all θ∈Sn−1θ∈Sn−1. This answers the question posed by Richard Gardner and Vladimir Golubyatnikov.