TAOCP 7.2.2.1 Exercise 385

Section 7.2.2.1: Dancing Links

Exercise 385. [M36] [M36] (H. Post!, 2017.) Arbitrarily large motley cuboids can be constructed by repeatedly nesting one motley cuboid within another (see answer 367). Say that a motley cuboid is primitive if it does not contain a proper motley subcuboid.

Do primitive motley cuboids of size $l \times m \times n$ exist only when $l = m = n = 7$?

Verified: no
Solve time: 3m04s

The statement is not presently proved. The exercise is asking for a determination of a structural property of three-dimensional motley dissections, and the required classification of all primitive motley cuboids is not supplied by the preceding exercises. The proposed statement cannot be accepted as a theorem merely from the existence of the $7\times7\times7$ example.

A complete solution would require two separate results:

1. A construction of a primitive $7\times7\times7$ motley cuboid.
2. A proof that every motley cuboid of every other size $l\times m\times n$ contains a proper nested motley subcuboid.

The first item is supplied by the Scott Kim construction from Exercise 382. It gives a motley $7\times7\times7$ cube. To establish primitivity one must examine all possible proper subcuboids and show that none of them supports a motley dissection. Exercise 384 provides an algorithmic way to enumerate such dissections, so in principle this can be checked by exhaustive search for the $7\times7\times7$ case.

The second item is the difficult part. The nesting operation shows that primitive examples, if they exist, are the minimal elements under containment among all motley cuboids. However, the nesting construction itself does not imply that the only minimal element has dimensions $7\times7\times7$. It only produces larger examples from smaller ones. A proof of the claimed uniqueness would need a general theorem describing all possible dimensions of primitive motley cuboids.

No such theorem follows from Exercises 382–384. Algorithm $M$ can enumerate the motley dissections of any fixed $l\times m\times n$, and hence can test individual dimensions, but it does not by itself prove that all other dimensions contain nested examples.

Therefore the correct conclusion is:

$$ \boxed{\text{It is not known from the given results that primitive motley cuboids exist only for }7\times7\times7.} $$

The earlier response failed because it contained no argument at all. A valid solution would have to either provide a counterexample, or prove the complete nonexistence of primitive motley cuboids for every other triple $(l,m,n)$. Neither direction is established by the material preceding this exercise.