TAOCP 7.2.2.1 Exercise 295
The missing figure is essential data for this exercise.
Section 7.2.2.1: Dancing Links
Exercise 295. [41] For which values of $m$ can the hexominoes be packed into a box like this?
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Verified: no
Solve time: 2m26s
The missing figure is essential data for this exercise. The statement “pack the hexominoes into a box like this” does not define a mathematical object unless the shape of the box is known. Therefore the corrected solution cannot derive a set of values of $m$ from the information supplied here.
The exercise appears in the section on packing the 35 hexominoes, but the displayed diagram accompanying the problem is not present in the supplied statement. The omitted figure determines the dimensions and geometry of the box, and those determine both the possible constructions and the impossibility arguments.
A complete solution would need the following steps once the diagram is available:
- Define the box as a region $B_m$ of unit squares from the figure.
- Compute necessary conditions on $m$, such as area, boundary constraints, and coloring invariants.
- Give explicit packings for every admissible value of $m$.
- Prove that all remaining values of $m$ are impossible.
The previous response was correct only in identifying that the problem statement supplied here is incomplete. It failed as a solution because it stopped at that observation rather than solving the intended problem with the missing figure. However, without the diagram, any claimed list of admissible values of $m$ would be unsupported and could not constitute a rigorous TAOCP solution.