Stage: 4 and 5 Challenge Level:
Remove three consecutive numbers from the set of natural numbers
from $1$ to $n$. The mean of the remaining numbers is $7.5$ . Find
$n$ and the numbers that were removed.
What if any three numbers are removed, not necessarily consecutive
Notes for students
A true problem solver is like a detective. When there is an obvious
or routine method of solution anyone can follow the usual steps and
get a solution. Such mathematical 'problems' don't even need an
intelligent human being, they can simply be fed to a computer to do
the work automatically.
When solving real problems you have to search like a detective for
all the information you can gather and then try to make sense of
what you have found out.
Some solutions can be found to this problem by trial and
improvement. That is just the start of the investigation! What
other methods can be used? Which is the best method? What can we
discover about the mathematical ideas involved and how can we use
these ideas to find out more? When we have some solutions can we
prove that there are no remaining undiscovered solutions? Can we
pose and solve similar problems? What generalisations can we
It appears at first sight that there is too little information
given about this problem so we have to find out all we can. The
problem involves triangle numbers and the arithmetic mean and we
know it is about whole numbers. We can construct and simplify some
Can we prove that $n$ cannot be very small or very large so we only
have to test a limited number of cases?
Can we use a spreadsheet or other computer program to help search
When we find some solutions can we be sure we have found them all?