## Description

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

noted:

• 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

for against

A 1 4

B 1

C 5 0 6

D 4

E 4 2 2 2

Task

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?