01202 398938

This pyramid of cannonballs appears at the foot of the Armada Monument on Plymouth Hoe. It consists of six layers: a single cannonball at the apex, resting on a square of 4 cannonballs, resting on a square of 9 cannonballs, and so on for a total of 6 layers.

A reasonably curious mathematician might wonder how many cannonballs there are in the entire pyramid (including the “hidden” ones you can’t see on the inside of the pyramid).

The answer is the sixth **Square Pyramidal number **– which I’ll call **$P_6$**

To find the total we could just go from layer to layer, adding as we go:

$P_6 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2$

$= 1 + 4 + 9 + 16 + 25 + 36$

$= 91$.

To put this another way:

The first Square Pyramidal Number consists of only the first layer, so 1 object.

Add the next layer and we have 5 objects in total.

The third layer adds another $3^2 = 9$ objects, making 14 in total.

The sequence continues: 1, 5, 14, 30, 55, 91, …

Counting layer by layer works ok for small pyramids like our n=6, but suppose you had a 45 layer pyramid, starting with one at the apex and continuing for 44 more layers (with $45^2 = 2025$ cannonballs on just the base layer). That’s a lot of balls! – too many to count layer by layer. So now what?

- Visit the Online Encyclopaedia of Integer Sequences (OEIS); type the first few terms in and the OEIS will not only recognise the sequence but will give you the first 45 terms. Magic! Why not go ahead and try it now?
- As the nth Square Pyrmaidal Number is the sum of the first n square numbers, you can use this formula for the sum of the first n square numbers:

The $\sum$ is apronounced **sigma** and means “add all the terms together”. The proof (by induction) of this formula is well-known to any student of A-level Further Maths.

When n=6 our formula gives $P_6 = \frac{1}{6} \times 6 \times 7 \times 13 = 91$ – the same answer we found earlier. Try setting n=45 and comparing the answer with the 45th term from the OEIS database.

Yes! Other related sequences include the Tetrahedral numbers (given by the sum of the first n triangular numbers), and also the pentagonal, hexagonal or even higher order pyramidal numbers. Head over to the OEIS and if you can figure out the first few terms, the database will give you the next few, plus some interesting facts.

**NOTIFY ME OF NEW POSTS BY EMAIL** (approx one a month):

- SQUARE PYRAMIDAL NUMBERS
- WHY DOES AREA OF CIRCLE = πr SQUARED?
- DUDENEY NUMBERS
- WHAT’S SPECIAL ABOUT THE NUMBER ELEVEN?
- MATHEMATICAL MUSCLES
- THE CAR WHEEL GAMES (RECEPTION TO YEAR 6)
- TOP THINGS FOR MATHEMATICIANS TO DO IN WALES
- POINTLESS PLASTICS BY NUMBERS
- BRITAIN’S GOT TALENT RUNNER-UP 2019 IS A MATHEMAGICIAN!
- STAR POLYGONS
- CIRCULAR REASONING: TOP TIPS FOR USING A COMPASS
- TRIANGULAR NUMBERS AND PYTHAGOREAN TRIPLES – A SURPRISING RELATIONSHIP
- FUN WITH OCTAGONS
- ADVENTURES IN THE FOURTH DIMENSION
- STRICTLY COME COUNTING
- WHY ARE THE STONES AT GIANTS CAUSEWAY HEXAGONAL?
- WHAT’S SPECIAL ABOUT THE NUMBER TWO?
- WHAT’S SPECIAL ABOUT THE NUMBER ONE?
- DOGARITHMS
- HOW MOST PEOPLE CAN BE “BETTER THAN AVERAGE”!
- WHY IS x USED FOR THE UNKNOWN IN ALGEBRA?
- 5 REASONS MATHS IS THE MOST IMPORTANT SUBJECT
- PARALLAX, PENTAPRISMS AND PHOTOGRAPHY
- 6 WAYS TO MAKE MATHS FUN
- THE TEN COMMANDMENTS OF MATHS
- SUPERSTARS OF MATHS – JOHN VENN
- SUPERSTARS OF MATHS – RENE DESCARTES
- HEXAHEDRA AND OTHER “HEX” WORDS
- WHICH IS BETTER: METRIC OR IMPERIAL?
- HOW MANY GIFTS IN TOTAL IN “THE TWELVE DAYS OF CHRISTMAS”?
- FUN WITH THE NEW POLYMER FIVE POUND NOTE
- HOUSE OF MATHS MAKES THE NATIONAL NEWS!
- FACTORS AND MULTIPLES
- SUPERSTARS OF MATHS – ISAAC NEWTON
- SUPERSTARS OF MATHS – LEONHARD EULER
- WHAT IS THE POINT OF ALGEBRA?