3.10 Necessity and sufficiency 117
found. Having the skill to know which pieces
of data are needed can save considerable time
and effort. Solving this type of problem does
not need particular mathematical skills – just
some clear and logical thinking.
I have a small collection of three types of old
coin. The collection contains 15 coins in
total. There are more pennies than half-
crowns and more half-crowns than guineas.
Which one of the following single pieces of
information would enable you to know exactly
how many of each type of coin there was?
A There are 4 more half-crowns than
guineas.
B There are 5 more pennies than
guineas.
C There are 3 more pennies than half-
crowns.
D There is one fewer penny than guineas
and half-crowns together.Activity
Commentary
In this problem, we are being asked to find
which of the options is sufficient (along with
the information we have already been given)
to solve the problem.
There are 12 ways that 15 can be
partitioned into three different numbers:
Guineas Half-crowns Pennies1 2 121 3 111 4 101 5 91 6 8Guineas Half-crowns Pennies2 3 102 4 92 5 82 6 73 4 83 5 74 5 6Of the options given, only C gives a unique
set. If there are 3 more pennies than half-
crowns, there must be 8 pennies, 5 half-crowns
and 2 guineas.
Why do the other options not work?• We have met a new type of problem where,
rather than being asked to find a solution,
we are asked to find what pieces of
information are necessary or sufficient to
solve it.
• We have also encountered problems where
we have to find a solution, but need to
identify an additional piece of information
which is necessary either to help us
with the method of solution or to choose
between different possible solutions.
• We have learned the meaning, in this
context, of the words ‘necessary’ and
‘sufficient’.
• We have seen various types of problem
which require extra data: some needing
mathematical solutions; some only
requiring us to establish a logical method
of solution.Summary