lean_workbook_plus_20501

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

For the surface area and volume to be numerically equal, we must have $abc=2(ab+bc+ca)$ where the edge lengths are $a$ , $b$ , and $c$ . Dividing through by $2abc$ yields $\frac{1}{2} = \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ . This allows for the use of a well known logarithm identity. Substituting the logs in and swapping the bases gives $\frac{1}{2} = \log_x{2} + \log_x{3} + \log_x{4} = \log_x{24}$ . From here it's clear the answer is $24^2=\boxed{576}$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_20501  (a b c : ℝ)
  (h₀ : 0 < a ∧ 0 < b ∧ 0 < c)
  (h₁ : a * b * c = 2 * (a * b + b * c + c * a))
  (h₂ : 0 < a ∧ 0 < b ∧ 0 < c) :
  (a * b * c)^2 = 576   :=  by sorry

The target drops x and every logarithm that determines the requested answer.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Restore the logarithmic definitions a=log₂x,b=log₃x,c=log₄x and conclude x=576, not(abc)²=576.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.