TAOCP 7.1.2 Exercise 57
In Figure (45), the seven-segment encoding assigns a distinct display pattern to each 4-bit input $(x_1x_2x_3x_4)_2$, corresponding to the hexadecimal digits $0$ through $15$.
Section 7.1.2: Boolean Evaluation
Exercise 57. [19] [19] What seven-segment "digits" are displayed when $(x_1x_2x_3x_4)_2 > 9$ in (45)?
Verified: yes
Solve time: 1m20s
Solution
In Figure (45), the seven-segment encoding assigns a distinct display pattern to each 4-bit input $(x_1x_2x_3x_4)_2$, corresponding to the hexadecimal digits $0$ through $15$. For inputs exceeding $9$, the encoding switches from decimal digits to alphabetic hexadecimal symbols.
For the values
$$ (x_1x_2x_3x_4)_2 \in {1010, 1011, 1100, 1101, 1110, 1111}, $$
that is, for integers $10$ through $15$, the displayed symbols are the standard seven-segment representations of hexadecimal digits.
The mapping given by (45) yields the following outputs:
$$ 10 \mapsto A,\quad 11 \mapsto b,\quad 12 \mapsto C,\quad 13 \mapsto d,\quad 14 \mapsto E,\quad 15 \mapsto F. $$
Thus the seven-segment digits displayed when $(x_1x_2x_3x_4)_2 > 9$ are
$$ \boxed{A,; b,; C,; d,; E,; F}. $$