There's a couple nice directions you could go here.
To discuss questions that are most similar to your example but more "authentic", you could look at any variety of constraint satisfaction problems. For example, Sudoku can be formulated as a bunch of constraints (each grid has exactly one of each number, no number appears multiple times, etc.) All of these can be formulated in propositional calculus. Another nice example is scheduling classrooms.
For something totally different, you could point out how ambiguous english can be and motivate logic as a precise system. My favorite way to do this is to choose some particularly hairy paragraph from a very old philosopher (e.g. some part of Pascal's wager), ask one of the students to read the whole thing quickly, and then ask the students if they understood anything. This isn't directly what you're asking for, I think, but it definitely helps motivate why mathematicians might care about logic to students who have read confusing prose (which is most of them).
For example, using the SMT solver Z3, this is code to solve Sudoku by defining logical formulae:
# define the puzzle as a 2D array here
from z3 import *
solution = Function('solution', IntSort(), IntSort(), IntSort(), BoolSort())
constraints = []
for i in range(len(puzzle)):
for j in range(len(puzzle)):
if puzzle[i][j] != 0:
# The result must match what we already know.
constraints.append(solution(i, j, puzzle[i][j]))
# Every cell must have some value
constraints.append(Or([solution(i, j, n) for n in range(1, 10)]))
#...but only one value
constraints.append(And([
Implies(
solution(i, j, n),
And([Not(solution(i, j, np)) for np in range(1, 10) if np != n])
)
for n in range(1, 10)]))
# Every row must have some value
constraints.append(And([
And([
Or([solution(i, j, n) for j in range(9)])
for i in range(9)])
for n in range(1, 10)]))
# Every column must have some value
constraints.append(And([
And([
Or([solution(i, j, n) for i in range(9)])
for j in range(9)])
for n in range(1, 10)]))
# Every box must have some value
constraints.append(And([
And([
Or([solution(i + boxi*3, j + boxj*3, n) for i in range(3) for j in range(3)])
for boxi in range(0, 2) for boxj in range(0, 2)])
for n in range(1, 10)]))
s = Solver()
s.add(constraints)
if s.check() == sat:
m = s.model()
r = [[ [n for n in range(1, 10) if is_true(m.evaluate(solution(i, j, n)))][0] for j in range(9)] for i in range(9)]
print_matrix(r)
else:
print "failed to solve"
