Equations of spheres
Problem 9.705 · easy
Find the equation of the sphere with center \( \displaystyle (-1, -1, -1) \) and radius \( \displaystyle 3 \), in standard form.
- A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
- \[ \left. \left(x + 1\right)^{2} + \left(y + 1\right)^{2} + \left(z + 1\right)^{2} \right|_{\substack{ x=2\\ y=-1\\ z=-1 }} = 9 \]A point at distance r along the x-direction is on it.✓ Proved
Answer \( \left(x + 1\right)^{2} + \left(y + 1\right)^{2} + \left(z + 1\right)^{2} = 9 \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | points at the claimed radius from the claimed center satisfy the equation |
Reviewers
gpt-oss:20b: fail (error) — The solution never derives the sphere’s equation; it only checks that a single point lies on a guessed equation, which is insufficient and misleading.qwen3.6:27b-mlx: pass — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius to derive the final answer. The verification step in line 2 is valid, as the point (2, -1, -1) is indeed at distance 3 from the center (-1, -1, -1).
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius to derive the final answer. The verification step in line 2 is valid, as the point (2, -1, -1) is indeed at distance 3 from the center (-1, -1, -1).gpt-oss:20b: fail (error) 2026-10-18 — The solution never derives the sphere’s equation; it only checks that a single point lies on a guessed equation, which is insufficient and misleading.qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution claims that the point (2, -1, -1) is on the sphere, but this point is at distance 3 from the center (-1, -1, -1) only if we consider the x-difference: 2 - (-1) = 3. Wait, the distance is sqrt((2 - -1)^2 + 0 + 0) = 3. So the point IS on the sphere. However, the equation line is labeled 'proved' by a CAS, but the sentence setup is just the standard formula. The issue is that the solution provides a single point check as 'equation, proved' which is not the equation of the sphere itself, but a verification step. The stated answer is correct. But the solution structure is weird. It defines the formula, then checks a point. It doesn't actually derive the equation. It just states the formula and checks a point. The 'equation' line is not the equation of the sphere, it's a substitution check. The problem asks for the equation. The solution does not explicitly state the final equation as a derived result, it just has a check. But the stated answer is provided separately. The review is of the solution lines. Line 1 is correct. Line 2 is a check, not the derivation. Is it wrong? It's misleading because it presents a point check as if it were part of the derivation of the equation, or implies that checking one point is sufficient to define the sphere. A sphere is defined by center and radius, not by one point. The solution fails to actually write down the equation derived from the formula in line 1. It jumps to a check. This is a 'misleading' or 'error' in logic. The student might think that finding one point is enough. Let's call it 'misleading' because it doesn't actually produce the equation, just verifies a point. Or 'error' because the solution is incomplete. The prompt asks to review the solution. The solution does not contain the equation of the sphere. It contains a definition and a check. The stated answer is correct, but the solution text is flawed. I will mark it as fail, severity error, because the solution does not actually solve the problem (find the equation), it just checks a point.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly substitute the given center (-1, -1, -1) and radius 3 into the standard equation formula. Instead, it performs an unrelated check of a specific point, which does not constitute a derivation of the equation.gpt-oss:20b: pass 2026-10-15
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/sphere_equation, checked 2026-10-18 with SymPy 1.14.0.