Virahāṅka–Fibonacci numbers, and the poetry-counting problem that produced them

These teaching notes are for members

What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

Create a free account Sign in

What to assume they know

Writing a number as a sum in several ways, and treating two orderings of the same parts as different
Odd and even as "can this be arranged in pairs": parity, needed for the pattern the section closes on
Reading a two-column table of cases and counts
The idea of a sequence and of its nth term, from The parity of a sum or product is fixed before you compute it
Adding two two-digit or three-digit numbers reliably
Counting a set by splitting it into cases that do not overlap

What they should be able to do

State the beat values of a short and a long syllable, and read a rhythm as a sum of ones and twos
List every way of writing a small number as an ordered sum of ones and twos, and count them
Explain why the count for n beats is the count for n − 1 added to the count for n − 2, using the opening-syllable split
Say why that split misses nothing and double-counts nothing
Continue the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55 and state a term several places further on
Find earlier terms of the sequence by subtracting instead of adding
Describe the repeating parity pattern of the sequence and use it to call the parity of a distant term without computing it
Place Piṅgala, Virahāṅka, Gopala, Hemachandra and Fibonacci in order, with approximate dates

Where it usually goes wrong

"The rule was spotted by looking at the numbers." It was derived from the counting problem. Handing over the rule and then verifying it on a list teaches the opposite lesson from the one on p.141.
"Rhythms that use the same syllables are the same rhythm." Order matters: 1 + 2 and 2 + 1 are two different three-beat rhythms and the table on p.140 counts them separately. A student who treats them as one gets 2, 3, 4, 5 instead of the sequence.
"Fibonacci discovered these numbers." He wrote about them in 1202 CE, after Virahāṅka by about five hundred years and after Piṅgala by far longer. The chapter says plainly that several Indian scholars had already written about them before him, and the joined name exists so this is not quietly forgotten.
"The sequence starts 1, 1, 2, 3, 5." As this chapter prints it, it starts 1, 2, 3, 5 — because the count of one-beat rhythms is 1 and the count of two-beat rhythms is 2. Both conventions are in use; the chapter's is fixed by what is being counted, and mixing them shifts every position by one.
"Every flower has a Virahāṅka number of petals." The chapter says petal counts generally fall this way, and shows three daisies. That is a tendency in some species, not a law of nature, and overstating it is the commonest way this material goes wrong on the internet.
"Going backwards down the sequence needs a new rule." It does not — the same relation read the other way is a subtraction, which is exactly what question 7 on p.144 asks for.
"Poetry is a decorative wrapper on a maths problem." The chapter's whole point is the reverse: the mathematics came out of a real question that prosodists needed answered, and the answer arrived as a Prakrit verse.

Questions to check understanding

Write a stated number as an ordered sum of ones and twos in all possible ways, and count them
Give the number of rhythms of a stated number of beats, and justify it by the opening-syllable split rather than by listing
Continue the sequence forward from two given consecutive terms, and backward
State the parity of a distant term, with the reason
Count the ways of climbing a staircase of n steps taking one or two at a time, and explain why the answer is the same as the rhythm count
Name the scholars associated with the sequence and place them in time
Explain why the sequence carries a joined name in this book
History-of-mathematics short answers on Virahāṅka and Piṅgala are the standard written form of this material

Examples worth working on the board

The sequence as printed (Part I, §6.4, p.139). 1, 2, 3, 5, 8, 13, 21, 34 and continuing. Later on p.141 it is extended to 55, and on p.142 to 89 with three further blanks for the reader.
Syllable durations (Part I, §6.4, p.139). A long syllable is held exactly twice as long as a short one; sung, that is two beats against one. The chapter names six languages whose poetry classifies every syllable this way: Sanskrit and Prakrit, and also Marathi, Malayalam, Tamil and Telugu.
Four eight-beat rhythms (Part I, §6.4, p.140). Printed as words. Writing S for a short syllable and L for a long one, the four are LLLL, then eight Ss, then S L L S L, then L L S S L. Checked against — on the page the long syllables are set in bold, which is worth keeping. Their arithmetic counterparts are printed just below: 2 + 2 + 2 + 2; eight 1s; 1 + 2 + 2 + 1 + 2; 2 + 2 + 1 + 1 + 2.
The small-case table (Part I, §6.4, p.140). Checked against the printed page. Three columns: the value of n, every way of writing it, and how many ways there are. Rows: n = 1 with one way; n = 2 with 1 + 1 and 2; n = 3 with three ways; n = 4 with five ways, all five written out.
The five-beat construction (Part I, §6.4, pp.140–141). Inputs: every 4-beat string with a 1 written in front of it, and every 3-beat string with a 2 written in front. Checked against the printed page: the resulting eight strings are printed in two blocks, five in the first and three in the second, which is exactly the 5 and the 3 being added. Do not flatten those two blocks into one list — the split is the argument.
Six beats (Part I, §6.4, p.141). Inputs: the counts for five beats and for four beats. The chapter then sets the reader to write all six-beat strings out and check.
The scholars (Part I, §6.4, p.141). Virahāṅka, a Prakrit scholar, gave this counting method around 700 CE, and set it out as a poem in Prakrit. Piṅgala, a Sanskrit scholar, came earlier, around 300 BCE, and Virahāṅka's work followed on from his. Gopala wrote about the numbers around 1135 CE and Hemachandra around 1150 CE. Fibonacci wrote about them in 1202 CE, roughly five hundred years after Virahāṅka. Use the dates as given; do not add biography the chapter does not supply.
The eight-beat answer (Part I, §6.4, p.141). Inputs: the sequence and the position wanted. The chapter reads the answer straight off, and the figure is highlighted in the printed line.
The parity question (Part I, §6.4, p.142). Inputs: the sequence and the question of whether the next term will be odd or even without adding. The pattern repeats with period three.
Daisies (Part I, §6.4, p.142). Checked against the printed page: three painted flower illustrations, not photographs, captioned with 13, 21 and 34 petals. The chapter's claim is hedged — a daisy's petal count is generally one of these numbers. Because the printed flowers are drawings, they cannot themselves settle a petal count; see Figures required.
The tabla drawing (Part I, §6.4, p.142). Three children round a pair of tabla, with a strip of coloured dashes beneath in short and long lengths. It is the rhythm idea returning as sound rather than as text.
The staircase (Part I, §6.5 Figure it Out, p.144, question 8). Eight steps, climbed one or two at a time. Inputs: the eight steps and the two allowed strides. This is the same count in different clothes and it is the best test that the argument, not the list, has been understood.
Neighbours in the sequence (Part I, §6.5 Figure it Out, p.144, question 7). Inputs: 987 and 1597, consecutive terms. The question asks for the next two and the previous two — subtraction going backwards is the point.

Figures to have open

A beat strip that can be filled with one-beat and two-beat blocks, and rebuilt as a sum of ones and twos. This is the central figure and everything else is secondary to it.
A growing enumeration panel for n = 1 up to n = 5, with the two prefix blocks visually separated.
A dated timeline with five marks. Standard schematic.
A daisy with countable petals. The textbook supplies painted illustrations here, not photographs, so it is not a route to a verifiable petal count: if the explanation wants the count checkable, a real photograph has to be sourced and cleared elsewhere. The printed illustrations may be redrawn, but then the claim is being asserted rather than shown.
A staircase of eight steps with a walker taking one or two at a time.
The chapter's tabla illustration is its own artwork; if drumming is used, redraw or use audio instead.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 6 "Number Play", §6.4 "Nature's Favourite Sequence: The Virahāṅka–Fibonacci Numbers!", pp.139–142 — the opening claim about Art, Science and Mathematics (p.139), the unnumbered block "Discovery of the Virahāṅka Numbers" with the syllable durations and the eight-beat question (pp.139–140), the small-case table and the five-beat construction (pp.140–141), the completeness argument, the scholars and the eight-beat answer (p.141), and the extension of the sequence with the parity question and the daisies (p.142)
Same part, §6.5 Figure it Out, p.144, questions 7, 8 and 9, which are the chapter's own assessment of this material
Same part, SUMMARY, p.145, fifth bullet, for the chapter's compressed statement that the sequence came out of the Arts
Solutions appendix bound after p.145 in the cached PDF (not part of the printed book), consulted for the intended answers to those three questions