Fun With Beads and Number Patterns 
 
Supurna Sinha, Raman Research Institute, Bangalore 
 
Nilanjana and Ratnamala were playing in the courtyard one lazy Sunday afternoon with beads and strings. Nilanjana said, "Ratna, that's a lovely pattern--what is it?". 
 
Ratnamala said, "I am making the wing of a butterfly with these coloured beads and strings." 
 
Nilanjana said, "I see a nice number pattern here as you go down from the top: the first layer has 1 bead, the next 2 beads, the third 3 beads and so on. (See Fig. 1). It is a sequence of consecutive natural numbers. 
 
How many beads? 
 
"How many beads do you want to put in the last  layer?" Ratnamala said, "I want to go all the way upto 10." 
 
Nilanjana said, "Shall we try to figure out how many beads you need  altogether to make such a wing pattern?" 
 
They set out to figure out the sum. Ratnamala took a piece of chalk and started decorating the courtyard with a chalk dot Kolam pattern  like the one that she had made with the beads she had strung together.  
 
She said, "Look Nilu, what a nice triangular pattern it makes!" 
 
Rectangles from Triangles  
 
Nilanjana exclaimed, "Yes, it does look beautiful. If we put two  such triangles together it will make a perfect rectangle". 
 
Ratnamala made such a rectangular pattern with two coloured chalk pieces (See Fig. 2). 
 
Nilanjana said, "Let's find out the total number of dots we have in the rectangle. The rectangle has a base of 4 dots and a height of 3 dots. So altogether there are 3 times 4 = 12 dots that form the rectangle. Since two triangles made up the rectangle, there are half as many, that is, six dots in the triangle. 

So the triangular patch of three layers has 6 dots. What about the triangle with 10 layers? Ratnamala said, "Before adding all the beads in ten layers, let me try out a few more sets with smaller numbers of dots. Let me try with a set of 4 layers."

She makes two such triangles and puts them together as before into a triangle with 5 dots on one side and 4 on the other (See Fig. 2). "Here," she says, "I have 4x5=20 dots making up the rectangle so there must be 20/2 = 10 dots in each triangular patch."

Nilanjana said, "Let me see: two 3-layer triangles formed a 4 by 3 rectangle. But two 4-layer triangles formed a 5 by 4 rectangle. In each case, the number of beads in each triangle is half that in the rectangle."

How many beads?

She thought for a while. "Let me see if I can find a general formula. If I have two n-layer triangles, they will form a (n+1) by n rectangle of area n(n+1). So the number of beads in the triangle will be n(n+1)/2."

"In other words, if we have natural numbers from 1 to n, the sum of all such numbers is  n(n+1)/2, which is the number of dots contained in a triangular patch of height n and base n+1. This comes from the fact that the number of dots in two such triangles is that in a rectangular patch of edge lengths n and n+1."

"So, if we have a butterfly wing with 1, 2, 3, ..., 10 beads strung on the first, second, third, ..., tenth layer, we need 10(10+1)/2 = 55 beads altogether."

Ratnamala said, "Let's finish making the butterfly wing and count all the beads on it". Ratnamala and Nilanjana put together a beautiful butterfly wing out of beads and string. Then they set out to count the beads on the wing. "One, two, three ..." Nilanjana exclaimed.  "It is indeed 55 altogether!"

Odd numbers

They were enjoying the Sunday afternoon silence staring at their work of art. Ratnamala broke the silence: "Hey, Nilu, I see some more interesting number patterns in the beads. As we go down the wing, and look at the triangular patterns of increasing size, I see there are 1, 3, 5, 7, 9 ... beads in every alternate layer. They form a series of odd numbers."

Nilanjana said, "That's a nice observation. I notice one more thing. If I add these numbers I can form a set of numbers which are perfect squares. I will show you what I mean.

1=1^2, 1+3=2^2, 1+3+5=3^2, 1+3+5+7=4^2, 1+3+5+7+9=5^2, 1+3+5+7+9+11=6^2 and so on."

Ratnamala said, "Wow, that's fun!"

Then they set out to make a beautiful Rangoli pattern on the courtyard.  Nilanjana drew a square and filled it up with white. Then Ratnamala drew three empty squares around the white square to form a block of 4 squares altogether. Nilanjana exclaimed, "This is exactly what I noticed. 1+3 = 4 = 2^2 (See Fig. 3).

They finished making the square pattern of two colours on alternate layers of the design. Then they modified the pattern a bit and changed the square blocks into flowers (See Fig. 4).

Nilanjana said, "That was fun. We learnt as much Art as Mathematics today." Ratnamala said, "Great! We should do this more often!"