There are two problems about missing scores.
1. A school held special examinations to decide which student in year 12 was best
overall in the subjects English, History, French, Mathematics, and Science. Five
students—Alan, Barbara, Charles, David, and Ellen—sat five papers each, one in
each subject. The top student in each paper was given 5 marks, the next student
4, and so on, the lowest scoring student receiving 1 mark. There were no ties in
any of the papers. After the marks had been allocated the following facts were
• Alan had an aggregate mark of 24.
• Charles had the same mark in four out of the five subjects.
• Ellen had topped Mathematics, and came third in Science.
• The students’ aggregate marks were in alphabetical order, and no two students
had the same aggregate.
2. Partway through a round-robin soccer tournament involving five teams, all official
match records were accidentally destroyed. The parts that could be entered
with certainty from memory are shown in the table below. Two points are given
for a win, one point for a draw, and zero points for a loss. Each team was supposed
to play each of the others once.
Team Played Won Lost Drawn Goals Goals Points
A 1 4
C 5 0 6
E 4 2 2 2
Answer the following questions in a way that convinces the reader that the answers
must be correct.
1. (School examinations) What was Barbara’s mark in Mathematics? Which students,
if any, obtained the same mark in at least four out of the five subjects?
2. (Soccer tournament) Has Team C played Team D yet? If yes, what was the score
of that game?