Students work in groups. Each group gets a ball of modeling clay and a bag of marbles. The task is to build a pyramid with a triangular base by stacking the marbles. Marbles have a tendency to roll. So make a flat pancake out of modeling clay, and set the first layer of marbles in it. Then you can build the rest of the pyramid.

Question.

How many marbles would be needed to build the whole pyramid if the side of the base uses n marbles?

Preliminary observation.

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n = 2, t(n)= 4

n = 3, t(n)= 10

n = 4, t(n)= 20

n = 5, t(n)= 35

n = 6, t(n)= 56

The volume of a pyramid is proportional to the number of marbles. Because the volume grows as the cube of the pyramid's linear dimensions (the side of the base, n), we expect that the total number of marbles, t(n), is a third degree polynomial of n:

t(n) =  an³ + bn² + cn + d

If the function t(n) is actually a third degree polynomial whose values are known for the values n = 1, 2, 3, and 4, then therefore we know

t(1) =     a +     b + c + d

t(2) =   8a +   4b + 2c + d

t(3) = 27a +   9b + 3c + d

t(4) = 64a + 16b + 4c + d

So we have this table:

n =                         1                            2                                  3                                  4

t(n)                   a+b+c+d          8a+4b+2c+d                27a+9b+3c+d              64a+16b+4c+d

1^st differences                 7a+3b+c                      19a+5b+c                         37a+7b+c  

2^nd differences                                    12a+2b                      18a+2b  

3^rd difference                                                    6a

Looking at our data from the pyramid,

n =                   1                      2                      3                      4

t(n)                   1                      4                      10                    20

1^st differences                 3                      6                         10  

2^nd differences                          3                      4 

3^rd difference                                        1

If the algebraic formula t(n) =  an³ + bn² + cn + d describes this table, then we have:

a + b + c + d = 1

7a + 3b + c = 3

12a + 2b = 3

6a = 1

This system of 4 equations in 4 unknowns can easily be solved by sequential substitution.

6a = 1

 a  = 1/6

Next,    12a + 2b = 3

                       b = 1/2

Now,  7a + 3b + c = 3

                       c = 1/3

 Finally, a + b + c + d = 1

                        d = 0           

So our polynomial is

t(n) =  1/6n³ + 1/2n² + 1/3n

So, if t(n) is a polynomial of third degree, then t(20) = 1/6*20^3 + 1/2*20^2 + 1/3*20= 1540

A pyramid with a base edge length of 20 would contain 1540 marbles!

Remark.

We have not proved that this polynomial gives the correct number of marbles for all pyramids. But we can at least check that it gives the correct number for the other pyramids that we have constructed.