TAOCP 7.2.2.2 Exercise 464
I cannot produce a rigorous solution for Exercise 7.
Section 7.2.2.2: Satisfiability
Exercise 464. ▶ [M25] $[M25]$ Transformations $\tau_1$ and $\tau_2$ don't change the text's example $10 \times 10$ matrix. Prove that they will never change any $10 \times 10$ matrix of sweep 3 that has $\nu X = 51$.
Verified: no
Solve time: 4m11s
I cannot produce a rigorous solution for Exercise 7.2.2.2.464 from the supplied context alone because the definitions of $\tau_1$, $\tau_2$, $\tau_3$, $\operatorname{sweep}(X)$, and $\nu X$ are not included.
The proof depends on the exact local conditions under which $\tau_1$ and $\tau_2$ modify a binary matrix. In particular, the statement
$\nu X=51,\qquad \operatorname{sweep}(X)=3$
does not determine the required argument unless the meaning of sweep and the actions of $\tau_1$ and $\tau_2$ are specified. Different plausible definitions of these transformations lead to different invariants and different proofs.
Please provide the part of Section 7.2.2.2 containing the definitions of $\operatorname{sweep}$ and $\tau_1,\tau_2,\tau_3$ (or the text immediately before Exercises 461–464), and I will write the complete textbook-style solution with the required proof.