### Why do this problem?

This
puzzle provides an interesting context which challenges pupils
to apply their knowledge of the properties of numbers. Pupils need
to work with various types of numbers at the same time and consider
their relationships to each other (e.g. primes, squares and
specific sets of multiples).

### Possible approach

Show a $3 \times 3$ grid
with six headings on the board, ask pupils to suggest numbers that
could fit into each of the nine segments (an easy start, but useful
revision of vocabulary).

The students (ideally
working in twos or threes) can then be set the challenge of filling
the $5 \times 5$ board with the available numbers.

There isn't a single
solution so students could display their different arrangements.
When a pupil/pair finishes allocating numbers to a grid, they
should record the grid headings and how many numbers they
placed.

The current "winning"
pupil's name could be on the board as a challenge, to be beaten; or
pupils could win points $10, 8, 6, 4, 2$ for each grid filled with
$25, 24, 23, 22, 21$ numbers respectively.

A concluding plenary
could ask pupils to share any insights and strategies that helped
them succeed at this task.

A teacher comments:

This is definitely one that needs them
to persevere. My class spent a full hour on this in groups and not
one group found a solution.

### Key questions

Which numbers are hard to
place?

Which intersections are
impossible?

Encourage pupils to pay
attention to the order in which they allocate numbers to cells -
recognising the key cells to fill, and the key numbers to
place.

### Possible extension

Teachers can adapt the task by changing the heading cards or by
asking students to create a new set of heading cards and a set of
numbers that make it possible to fill the board. Students could
then swap their new puzzles.

Is it possible to create a puzzle that can be filled with $25$
consecutive numbers?

### Possible support

Some pupils could be
given a larger range of numbers to choose from, or offered a
smaller grid and appropriately restricted numbers - this could work
with pupils choosing from the full set of $10$ categories, or with
an adapted set.

Teachers may be interested in Gillian Hatch's article

Using Games in the Classroom in which she analyses what goes on
when mathematical games are used as a pedagogic device.

Handouts for teachers are available here (

word
document,

pdf
document), with the problem on one side and the notes on the
other.