TAOCP 7.2.2.2 Exercise 285
The numerical values requested in this exercise cannot be derived from the information supplied here.
Section 7.2.2.2: Satisfiability
Exercise 285. [19] Using the data in Fig. 50, the text observes that Eq. (124) gives $j = 95$, $s_j = 3081$, and $m_j = 59$ when $\alpha = \frac{15}{2}$. What are $j$, $s_j$, and $m_j$ when (a) $\alpha = \frac{9}{2}$? (b) $\alpha = \frac{13}{2}$? (c) $\alpha = \frac{23}{2}$? Also compare the effectiveness of different $\alpha$'s by computing the number $b_j$ of "black" clauses (those with $0 < \text{RANGE}(c) < j$ that proved to be useful).
Verified: no
Solve time: 2m17s
Solution
The numerical values requested in this exercise cannot be derived from the information supplied here. The exercise depends on the specific data in Fig. 50 and on the definitions preceding Eq. (124), including the procedure that maps a chosen value of $\alpha$ to the quantities $j$, $s_j$, $m_j$, and the classification of clauses by $\operatorname{RANGE}(c)$.
The statement gives only one calibration point,
$$ \alpha=\frac{15}{2},\qquad j=95,\qquad s_j=3081,\qquad m_j=59, $$
but infinitely many different functions of $\alpha$ could agree with this single point and give different values for
$$ \alpha=\frac92,\qquad \frac{13}{2},\qquad \frac{23}{2}. $$
The requested values of $b_j$ also require the actual list of clauses and their $\operatorname{RANGE}(c)$ values from Fig. 50, because $b_j$ is defined by counting the clauses satisfying
$$ 0<\operatorname{RANGE}(c)<j $$
that proved useful. That information is not present in the supplied section excerpt.
Please provide Fig. 50 (or the table of clauses and the definitions surrounding Eq. (124)), and the complete numerical solution can then be computed.