Figurate Numbers, Supermarkets, and Pascal TrianglesAuthor(s): Arnall RichardsSource: Mathematics in School, Vol. 4, No. 5 (Sep., 1975), pp. 2-4Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211424 .

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]FigLurate nmers superm by Arnall Richards, North Walsham High School, triagles Norfolk

Figurate numbers have been around for a long time. The Greeks were fascinated by them-probably because, like the Romans, they were badly hindered in much of their arithmetic by a clumsy notation, and a notation-free geometric approach offered a partial salvation. But are they of importance today? / believe that they are, and hope to show in this article one or two ways in which they help our understanding of mathematics.

Although most teachers in training learn something of the two simplest kinds of figurate number (triangular and square num- bers) they are often unsure of the formal rules for constructing figurate numbers in general. This point is underlined by three recent articles in Mathematics in School (1) (2) (8) which use the term "hexagonal number" incorrectly. All figurate numbers are the partial sums of arithmetic progressions (see Table) and much of their importance lies in the light they throw on methods of summing Arithmetic Progressions and other series. A useful idea in this connection is that of a gnomon; although strictly the name gnomon should be reserved for the L-shaped pieces which are used to build up the square numbers I shall use it idosyncratically to mean any shape which is used to build up a sequence of figurate numbers. Figure 1 shows the method of construction applied to the first few triangular, square, pent- agonal and hexagonal numbers.

Figure 1

T3=1+2+3 S3=1+3+5

P3=1+4 +7 H3=1 +5+9

Triangular numbers These are the sums of the first one, two, three., whole num- bers. The nth triangular number is therefore the sum of the first n whole numbers, and if a way can be found to write down a formula for the nth triangular number (Tn for short) then clearly we have a formula for finding the sum of the first n whole numbers. It turns out that such a formula is surprisingly easy to find. If two Tn's are placed "head to tail" they form a rectangle measuring n units one way and n+l units the other (figure 2). Hence 2Tn = n(n+l), from which we deduce that Tn = n(n+1).

2

Figure 2 0 o 0 o o x 00 0 0

0T x x

0 0 0 x x x

0 0 x x x x o x x x x x

Showing Tn+Tn= n (n+1)1in the case n=5

A second relation between different kinds of figurate numbers is shown in figure 3; namely, Tn+Tn-1 = Sn (the nth square number). We shall return to this important relation later, when considering the sums of square numbers.

Figure 3

Show Tn+Tn-1= Sn in the case n= 5

Square numbers It is apparent from figure 1 that the square numbers are formed by adding the consecutive odd numbers: 12 = 1, 22 = 1+3, 32 = 1+3+5, etc. We deduce that the sum of the first n odd numbers is n2. A slightly less obvious deduction is that every odd number (except unity) will appear as a member of a Pytha- gorean triad; this follows from the fact that every odd number takes its proper turn as a gnomon, including the odd squares 9, 25, 49, ...; and that the other two members of the triad will be consecutive whole numbers. (We must not, however, jump to the conclusion that every primitive triad can be formed in this way; 8, 15, 17 provides a counter-example.)

Pentagonal & Hexagonal numbers

These are of little special interest. Formulas can be found for

Pn and Hn (and hence for the A.P.s from which they are derived) by noting that Pn = Tn+2Tn--1 = 1)+(n-1)n = iin(3n-1) and

Hn = Tn+3Tn-1 = In(n+1)-+-(n-1)n = n(2n-1). (See figure 4)

Supermarkets & Cannonballs An interesting problem is that of determining how many tins of baked beans (or cannonballs!) are contained in a given pyramidal stack. The number will, of course, depend on the type of

pyramid chosen, and its dimensions (in terms of the number of tins to a side). If the pyramid is of triangular form (figure 5) decreasing uniformly to a single tin at the apex then we have to find 1+3+6+ ... +ln(n+l). Figure 6 shows how six such pyramids can be used to form a cuboid measuring n units by n+1 units by n+2 units. Hence the required sum is 6n(n+l)(n+2).

If the pyramid is square we have to find 12+22+3'+ +n2 This is most easily done by dividing each square layer into two triangular layers (sinceSn = Tn+Tn-1 ); the wholesquare pyramid' can thus be split into two triangular pyramids, one of side n, the other of side n-l. (See figure 7.) Using the result already ob- tained for triangular pyramids we deduce that the number of

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arkets, and pascal

This rectangular pyramid is made by combining two triangular pyramids 2(T1+T2+T3) = 2TI+2T2+2T3.

This cuboid is made by combining three rectangular pyramids 3(2T1+2T2+2T3) = 6T1+6T2+6T3 = 6(T1+T2+T3).

items in a square pyramid is ,n(n+l)(n+2)+-(n-1)n(n+l) = n(n+l)(2n+l). Both of these results can, of course, be obtained by normal algebraic manipulation; but the usual procedures are less intuitively obvious and involve heavier working than those outlined above.

Pascal triangles It is well known that the coefficients of the terms in the expan-

sion of (a+b)n form a regular pattern, usually displayed as "Pascal's triangle" (figure 8, which shows the method of con- struction). Once it is understood how the successive lines of the triangle are obtained it is easy to see why one sloping line should yield the triangular numbers (figure 9 shows T4 = 10 = 1+2+3+4) and the next one the triangular-pyramid numbers (figure 10 shows the fourth such number = 20 = 1+3+6+10).

3

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Figure 8

1

1%

61 I 1

6 15 20 15 6

Figure 9

1 2 1 2 1

1 3 +

)3 1

1 4 6 4 1

1 5 10 0 5 1

1 6 15 20 15 6 1

This sloping line gives the triangular numbers

Figure 10

1 1

1 2 1

+ 3 3 1

1 41 6 4 1

+33 0 10 5

6 15 15 6 1

Triangular pyramid numbers

Figure 11

1 2

1 3 2

1 4 5 2

1 5 9 7 2

1 6 14 16 9 2

1 7 20 30 25

Square numbers Square pyramid numbers

4

If we now return to figure 1 and look again at the gnomons used for the triangular numbers we can see that they are in-

creasing one at a time, being in fact the natural numbers 1, 2, 3, .... The gnomons for the square numbers increase two at a

time (1, 3, 5, ....), those for the pentagonal numbers three at a time (1, 4, 7, .. .) and so on. This suggests the modified Pascal triangles shown in figures 11 and 12 which show the square (and square-pyramid) numbers, and perntagonal (and pentagonal- pyramid) numbers respectively. The method of construction is simple and general: write units down the left-hand edge of the triangle and the appropriate differences down the right-hand edge.

Figure 12

1 3

1 4 3

S5 7ee

3

1 6 12 10 3

1 7 18 22 13 3

1 8 25 40 35 18 3

Pentagonal numbers Pentagonal pyramid numbers

Conclusion These brief notes are not intended to be the last word on

figurate numbers. A search through back issues of Mathematics in School will reveal many little problems based on their proper- ties which I have not mentioned. One last thought: if successive lines of a Pascal triangle represent respectively a zero-dimen- sional difference, a one-dimensional gnomon, a two-dimensional figurate number, and a three-dimensional pyramid number, what will the next one represent? and the one after that? 0

Table of Figurate Numbers

Name

Triangular Square Pentagonal Hexagonal

A.P. from which derived

1+2+3+4+... 1+3+5+7+... 1+4+7+10+... 1+5+9+13+...

First four numbers

1,3,6, 10 1,4,9, 16 1,5, 12,22, 1,6, 15,28

General term

yn(n+l) 2 n

-n(3n-1) n(2n-1)

References

All references are to articles which have appeared in Mathe- matics in School

1. Atherton, Roy. 2. Clements, M. G. 3. Eperson, D. B. 4. -

5. Holt, Michael. 6. -

7. Whittaker, Dora.

8. Powell, Martin R.

Hexagonal numbers in context. November 1974. Hexagonal numbers. May 1974. Number patterns. March 1972. Puzzles, Pastimes and Problems. March 1972; May 1972; November 1972; March 1974. ABC of mathematicians (Cauchy). May 1972. Tangling with squares. September 1974. That rare phenomenon-the gifted pupil. March 1974. More Hexagonal Numbers. May 1975.

Editor's Footnote: Another reader, Frank Tapson, has pointed out that the articles (1), (2) and (8) above should refer to "centred hexagonal numbers", first studied in 1739, as opposed to 500 BC for proper hexagonal numbers. A recent reference is to be found in Martin Gardner's column in The Scientific American for July 1974.

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