Why do this problem?
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).
This printable worksheet may be useful: The Factors and Multiples Puzzle
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.
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.
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?
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.