Proving the Theorem of Nicomachus


*This course is intended for lower elementary students


Class Schedule

Mondays 5:30-6:20 pm PST

March 11, March 18, March 25, April 1, April 8


Note: Classes are listed in PST, click below to convert time to your time zone.

Course Overview

Small Group Advanced Math

This 5-week course will be a maximum of 3 students in this amazing opportunity to work with Master Teacher, Dr. Peter Koehler.

Weaving Strands of the Times Tables into Gnomons and Cube Numbers to Prove the Theorem by Nicomachus

We’ll play and experiment hands-on with the geometric and mathematical structures made from interlocking blocks to discover some  unsuspected and surprising properties of the whole numbers. We will look for triangle numbers in our visualization of the multiplication square, see how they are hidden within it and learn why the squares of the triangle numbers are related to the sum of the cube numbers.  We will create a rich tapestry of profound patterns in search of our goal, which is to discover the visual proof of the Theorem by Nicomachus, to see how everything falls into place and makes sense.

The Theorem by Nicomachus: ‘the square of the nth triangle number or T(n)^2 is equal to the sum of the first n consecutive cube numbers starting with 1‘


We begin by making a 9x9 multiplication square on paper making several copies to look for patterns which we highlight.  I try to follow each student’s lead in the patterns they discover. Using the interlocking blocks we then build the first small multiplication squares and their L-shaped gnomons.  A traditional multiplication square (M square) is a useful mnemonic device to learn and remember the basic multiplication facts. 

(Click here to view course description with visuals)

Materials Required: 

Student Experience

Pattern Seeker




Your Teacher: Dr. Peter Koehler

Peter Koehler holds a PhD in theoretical and elementary particle physics from Royal Holloway College, University of London; a master’s degree from Imperial College of Science, Technology and Medicine, London; and carried out post-doc studies in the theory group at Stanford Linear Accelerator Center before becoming a math enrichment teacher at Nueva, where he has been teaching for over 20 years. At Nueva, Peter has become particularly interested in encouraging and fostering mathematical creativity in his students and was awarded a fellowship from Johns Hopkins University for excellence in teaching in 2012. He enjoys showing his students the surprising ways in which math can be used to describe aspects of the natural world. Inspired by the work of the Pythagoreans, he has developed an approach to elementary math teaching where the students use colored interlinking blocks and follow a few simple rules to visualize numbers; look for patterns, shapes, and sequences; make their own mathematical creations; and develop a sense of the more general principles of mathematics. He has found that this approach stimulates interest and enthusiasm for math, is a great motivator, and can spark mathematical creativity, originality, and a joy in the subject, and can lead to more intriguing and advanced aspects of math.

Peter has been a regular presenter at the Nueva ILC conferences and will be presenting a paper at the 11th International Conference on Mathematical Creativity and Giftedness in Hamburg, Germany, in 2019. He has taught independent enrichment programs at several Bay Area schools and the University of Santa Cruz extension. A painter in his spare time, Peter has run visual arts summer camps throughout the Bay Area for the past 25 years. He has also written plays for children.