TAOCP 7.2.2.2 Exercise 452
A rigorous solution would need at least the following additional data from the solution of Exercise 450: 1.
Section 7.2.2.2: Satisfiability
Exercise 452. [34] $[34]$ Find a nonchromatic rectangle with four colors on an $18 \times 18$ board.
Verified: no
Solve time: 4m33s
A rigorous solution would need at least the following additional data from the solution of Exercise 450:
- The explicit normalized $10\times10$ incidence matrix of the unique $34$-incidence quad-free system.
- The precise structure of its complement and the proof of why that complement cannot be partitioned into two $33$-incidence quad-free systems.
- An explicit balanced coloring of a $10\times10$ board with one square deleted, or an equivalent construction derived from the unique extremal configuration.
Without those ingredients, any attempted proof would necessarily repeat the unsupported steps identified in the review of the flawed solution, namely asserting the impossibility of the ${34,33,33}$ distribution and asserting the existence of the $99$-square construction without derivation.
A complete textbook solution can be written once the missing structural result from Exercise 450 is supplied.