TAOCP 7.2.2.1 Exercise 375

Section 7.2.2.1: Dancing Links

Exercise 375. [M29] [M29] Among all the incomparable dissections of order (a) seven and (b) eight, restricted to integer sizes, find the rectangles with the smallest possible semiperimeter (height plus width). Also find the smallest possible squares that have incomparable dissections in integers. Hint: Show that there are $2^l$ potential ways to fix the $b$'s with the $u$'s, preserving their order; and find the smallest semiperimeter for each of those cases.

Verified: no
Solve time: 3m22s

A complete corrected solution cannot be written from the information supplied in the prompt. The previous response was incomplete as a response to the exercise, but its central observation was mathematically valid: Exercise 7.2.2.1.375 is not self-contained.

The exercise depends essentially on the constructions and notation of the preceding exercises in Section 7.2.2.1, especially:

1. The labelled $4\times4$ motley reduction from Exercise 365, which supplies the seven-region order-$7$ case.
2. The characterization in Exercise 374(e), which converts a dissection into linear inequalities in the variables $x_i,y_i$.
3. The enumeration of the possible assignments of the $b$'s to the $u$'s in the order-$8$ case.

Without those data, the numerical minima requested in the exercise cannot be derived. In particular, the missing information is not merely explanatory background. It determines the actual systems of inequalities whose integer solutions give the extremal rectangles.

The required proof has the following structure.

For a fixed reduction, let the region widths and heights be expressed as

$$ w_i=\sum_j a_{ij}x_j,\qquad h_i=\sum_k c_{ik}y_k, $$

where the $x_j$'s and $y_k$'s are the positive integer strip widths and heights of the reduction. The incomparable condition requires

$$ w_1<w_2<\cdots <w_t $$

and

$$ h_1>h_2>\cdots >h_t . $$

For each possible ordering of the strip variables, the inequalities split into two independent systems:

$$ A x>0,\qquad C y>0 . $$

The smallest rectangle associated with that case is obtained by minimizing

$$ H+W, $$

where

$$ H=\sum_k y_k,\qquad W=\sum_j x_j, $$

subject to positivity and integrality.

The hint refers to the fact that, for $l$ relevant comparisons, each comparison can be resolved in two possible ways. Hence the possible assignments of the $b$'s to the $u$'s while preserving order number

$$ 2^l . $$

Every one of these cases must be solved, and the smallest resulting value of $H+W$ is the optimum. The final step is then to check whether $H=W$ is possible; the least such value gives the smallest square.

The numerical answers require the actual matrices $A$ and $C$, or equivalently the labelled dissections from the earlier exercises. They are not present in the problem statement or in the proposed solution. Supplying numerical semiperimeters or square sizes without those data would amount to inventing the missing dissections rather than proving the exercise.

Therefore the previous solution should not have been presented as a completed solution, but the reviewer’s demand for concrete numerical values cannot be satisfied from the supplied material alone. The missing pages or figures from the relevant part of Section 7.2.2.1 are necessary before a rigorous textbook-style solution with the requested minima can be produced.