lean_workbook_plus_21805

⚠️ 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.

Let $S=(-4,\ldots,4)^6$ . If $(a,b),(c,d),(e,f)$ are the three points, then the area of the triangle formed is $\frac{1}{4}(ad-bc+cf-de+eb-af)^2.$ Summing up over all of $S$ and using symmetry, we have $\sum_S{K^2}=\frac{6}{4}\sum_S{a^2d^2}+t_1\sum_S{abcd}+t_2\sum_S{a^2df}$ for some constants $t_1,t_2$ . But $2\sum_S{abcd}=\sum_S{abcd}+\sum_S{abc(-d)}=0$ (if $d\ne0$ , we can pair up the terms, and if $d=0$ , the expression is not affected) and similarly, $2\sum_S{a^2df}=\sum_S{a^2df}+\sum_S{a^2(-d)f}=0.$ So $E=\frac{\frac{6}{4}\sum_S{a^2d^2}}{9^6}=\frac{\frac{6}{4}9^4(\sum_{i=-4}^{4}i^2)^2}{9^6}=\boxed{\frac{200}{3}}.$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_21805  (a b c d e f : ℤ)
  (h₀ : -4 ≤ a ∧ a ≤ 4)
  (h₁ : -4 ≤ b ∧ b ≤ 4)
  (h₂ : -4 ≤ c ∧ c ≤ 4)
  (h₃ : -4 ≤ d ∧ d ≤ 4)
  (h₄ : -4 ≤ e ∧ e ≤ 4)
  (h₅ : -4 ≤ f ∧ f ≤ 4)
  (h₆ : a ≠ b)
  (h₇ : a ≠ c)
  (h₈ : a ≠ d)
  (h₉ : a ≠ e)
  (h₁₀ : a ≠ f)
  (h₁₁ : b ≠ c)
  (h₁₂ : b ≠ d)
  (h₁₃ : b ≠ e)
  (h₁₄ : b ≠ f)
  (h₁₅ : c ≠ d)
  (h₁₆ : c ≠ e)
  (h₁₇ : c ≠ f)
  (h₁₈ : d ≠ e)
  (h₁₉ : d ≠ f)
  (h₂₀ : e ≠ f) :
  1 / 4 * (a * d - b * c + c * f - d * e + e * b - f * a)^2 = 200 / 3   :=  by sorry

Take the rational/real average of squared triangle area over all9^6 coordinate tuples in{−4,…,4}^6 and conclude that average is200/3; do not assert every tuple has that value or require all six coordinates distinct.

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

Proposed corrected statement

Take the rational/real average of squared triangle area over all9^6 coordinate tuples in{−4,…,4}^6 and conclude that average is200/3; do not assert every tuple has that value or require all six coordinates distinct.

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.