This problem offers an excellent opportunity for students to practise visualisation and apply an idea normally only used in 2D geometry to a 3D case. Learners will have to consider carefully how to communicate their methods for testing combinations and that they have considered all possibilities.
This printable worksheet may be useful: Tet Trouble.
In silence, write three lengths on the board (for example 3 units, 6 units, 7units) and accurately draw a triangle with sides of corresponding lengths. You could use a dynamic geometry package to do this.
Do it again with three more lengths.
And again but instead of drawing the triangle put a question mark. After some thinking time, encourage a member of the group to come up and draw the triangle.
Finally, list three lengths that will not work followed by a question mark and after time has been taken to realise the impossibility, discuss why this is the case as a group.
Now pose the problem.
Working in small groups the challenge will be to employ systematic approaches as well as applying the triangle inequality.
Take opportunities to pull together different ideas for recording, including the use of nets and working systematically.
Or there is the problem : Sliced . This is a challenging next step in this kind of visualisation.