TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

It's quite challenging for many students to understand that a tetrahedron can indeed be formed in the way described. It's recommended to have scissors, paper, and rulers available for students to build accurate models.

Prerequisites

A course in high school geometry is sufficient. In particular, the students should have been introduced to three dimensional objects. Familiarity with the volume formula for a tetrahedron given height and base would help.

Calculus can be used, but is certainly not necessary.

Time allotment - time management

In class, you could spend between 30 - 50 minutes on this problem. Students would then be expected to complete and write-up solutions as homework with at least 2 nights after the class work to turn it in.

Expectations

Students have a difficult time visualizing this problem without the help of a model.

Students will have a difficult time determining the height of the pyramid -- as opposed to the height of a side of the pyramid (which some may need reviewed.)

Be prepared to offer hints.

Future payoffs

Improves visualization skills.

Encourages creative thinking.

Extensions

Note that the volume of the pyramid was integer valued. Check that the following integer triples (other than (11, 20, 21)) also yield a tetrahedron of integer volume:
(33, 65, 72); (69, 91, 100); (21, 99, 100).
Can you find other integer triples that yield a tetrahedron of integer volume?

Are there infinitely or finitely many such integer triples? (If finite --- can you say how many?)

These last two questions are only for the strong willed students who really got involved with the original question and are yearning for more!

References and Sources

USA Math Talent Search problem: Year 3, Round 3, problem 5.
Contact George Berzsenyi, Rose-Hulman Institute of Technology,
5500 Wabash Avenue, Terre Haute, IN 47803. (812) 877-8474.