The Nonlinear Classroom

1000 for the 10 by 10 grid

n^3 for the n \times n grid ( n \geq 1 ).

One solution strategy is to break the square up into columns, get the total of each column, and then add the totals together (55 + 65 + 75 + … + 145 = 1000). Notice that the sum of each column is 10 more than the column to its left, exactly because each number is 1 more than the number to its left, and each column contains ten entries.

A similar strategy is to group the rows. The non-conformists might group diagonals (either top left to bottom right, or top right to bottom left, both of which have nice structure).

Below are some fun observations and detours that I made while working on this problem. What paths did you take? Leave a comment and let me know what you found interesting about this problem.

A Recursive Perspective

Instead of a static square of numbers I imagined a dynamic pattern growing inside of the square – a nested sequence of small squares where a recursive rule governs the changes from each square to the next. This rule is shown visually in the image below. 

Now add up the numbers in each of the small grids. The sums are: 1, 8, 27, 64 which are the perfect cubes: 1^3, 2^3, 3^3, 4^3, \ldots The original puzzle is for the 10 by 10 case, so we conjecture that the sum is 10 cubed, or 1,000.

Cubes from triangular numbers

Adding the first row (or column) of the 10 \times 10 square gives the 1 + 2 + 3 + \cdots + 10 , also known as the 10th triangular number which is quickly computed by {11 \choose 2 } = 55. .

For the 10 \times 10 \times 10 case, the fact that each row sum is 10 more than the row above it gives the total sum in terms of triangular numbers: \begin{aligned}10^3 &= {11 \choose 2} + {11 \choose 2} + 10 + {11 \choose 2} + 20 + \cdots + {11 \choose 2} + 90 \\ & = 10 \cdot {11 \choose 2} + (10 + 20 + \cdots + 90) \\ &=10 \left( {11 \choose 2} + (1 + 2 + \cdots +9) \right) \\ &= 10 \cdot \left({11 \choose 2} + {10 \choose 2}\right). \end{aligned}

The formula for the n \times n \times n cube is thus,

\begin{aligned} n^3 = n \cdot ( {n + 1 \choose 2} + {n \choose 2} ), \end{aligned}
which as an algebraic representation of the pattern strikes me as quite pretty.

Discovery Learning Worksheet

Click Here. To edit the document and make it your own, open the File menu and select “Make a Copy”.

I use the worksheet to introduce cubic functions and recursion (which is fundamental to how I teach polynomial functions).

Manipulatives for Class

It’s easy to make this puzzle a physical manipulative using blocks. The height of the block tower below is given by the number in each grid square.

Rearranging the blocks from a staircase pattern to a cube provides a beautiful visual proof of the problem.

Also, this perspective may be useful for introducing functions of two variables, heat maps, or topographical maps.

Jo Boaler Alert!

Jo Boaler popularized the visual patterns “Stairs to Squares” and “Growing Shapes“, which I’ve reproduced below. If you already use these visual patterns while teaching quadratic functions, then it is powerful pedagogy to make the leap from 2D to 3D (from degree 2 polynomials to degree 3 polynomials).

Notice how the two dimensional patterns fit nicely inside of the 3D cubic tower:

Algebra

It might be a valuable exercise in algebra symbol manipulation and modeling to take the different decompositions of the grid that students find (by rows, columns, diagonals, etc.) and match them to the corresponding decomposition of the function y = x^3 .

Summary

There is a lot that your students can learn from this simple counting problem. While I use it as an introduction to cubic growth and cubic functions, I’m sure it can be adapted for many different levels of math – it makes for a great practice in numeracy and problem solving.

Breaking a large problem into small, manageable pieces is a classic problem solving skill that students need to practice. The fact that the smaller pieces nest recursively makes the pattern fun to play with, is used widely in the “real world” as a programming technique, and is a signature property of polynomial functions.

I found the problem for the 10 by 10 grid in Alex Bellos’ Guardian Puzzle series here, although I imagine this pattern is pretty old. Anyone know the history?