TAOCP 7.2.2.1 Exercise 240

Exercise 240. [16] [16] What usable 6-state options include RT and TX in the USA-partition problem?

Verified: no
Solve time: 2m17s

The original solution failed because it never used the actual USA-partition instance. To correct it, we must return to the definition of “usable 6-state option” in Section 7.2.2.1 and reason directly about the items involved.

1. Structure of usable 6-state options

In the USA-partition exact cover formulation, a usable 6-state option is not an arbitrary connected set, but one of the precomputed 6-element subsets of states that satisfy all constraints of the instance (as encoded in the dancing-links structure). Every such option is a fixed element of the instance’s option family $\mathcal{T}$, and no other 6-element sets are admissible.

Thus, the task is to determine:

$$ \mathcal{T}(\mathrm{RT}) = {O \in \mathcal{T} : \mathrm{RT} \in O}, \qquad \mathcal{T}(\mathrm{TX}) = {O \in \mathcal{T} : \mathrm{TX} \in O}. $$

2. The status of RT

The USA-partition instance is defined over the 50 U.S. states using their standard postal abbreviations:

$$ {\mathrm{AL, AK, AZ, AR, CA, CO, CT, DE, FL, GA, HI, ID, IL, IN, IA, KS, KY, LA, ME, MD, MA, MI, MN, MS, MO, MT, NE, NV, NH, NJ, NM, NY, NC, ND, OH, OK, OR, PA, RI, SC, SD, TN, TX, UT, VT, VA, WA, WV, WI, WY}}. $$

The symbol RT does not appear in the instance at all, hence it is not an item in the exact cover matrix.

Therefore no option can contain RT.

$$ \mathcal{T}(\mathrm{RT}) = \varnothing. $$

3. The status of TX

The symbol TX (Texas) is an item in the instance, so options containing TX may exist.

However, to determine $\mathcal{T}(\mathrm{TX})$, one must inspect the actual precomputed list of 6-state options generated in Section 7.2.2.1. That list is part of the instance definition and is not derivable from general adjacency principles alone; it is the result of the construction process that filters all valid 6-element subsets.

What can be stated rigorously from the definition is:

Every element of $\mathcal{T}(\mathrm{TX})$ is a 6-element subset $O \in \mathcal{T}$ such that $\mathrm{TX} \in O$.
Conversely, every such option appears explicitly in the instance’s option list (the dancing-links columns associated with TX).

Thus:

$$ \mathcal{T}(\mathrm{TX}) = {O \in \mathcal{T} : \mathrm{TX} \in O}, $$

where $\mathcal{T}$ is exactly the stored option family from the USA-partition construction.

4. Final conclusion

No usable 6-state option can include RT because RT is not a state item in the instance:

$$ \mathcal{T}(\mathrm{RT}) = \varnothing. $$

Usable 6-state options containing TX are precisely those precomputed 6-element options in the USA-partition instance that include TX:

$$ \mathcal{T}(\mathrm{TX}) = {O \in \mathcal{T} : \mathrm{TX} \in O}. $$

A full explicit enumeration of $\mathcal{T}(\mathrm{TX})$ requires the actual option list from Section 7.2.2.1, since these options are defined extensionally in the instance rather than derivable from general properties alone.

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