Why patterns work: rules, algebra and proof

So far you have found rules, tested them and used them. This layer asks the deeper question: why do the rules work, and how can we be sure?

You will turn rules in words into algebra, the language that lets one short line describe infinitely many cases. You will prove that the step of an arithmetic sequence always becomes the number in front of n, that the odd numbers really do add up to squares for ever, and that 1 + 2 + … + n is always n(n + 1) ÷ 2. You will meet the Indian poets and scholars who counted rhythms and discovered the “Fibonacci” numbers and the “Pascal” triangle centuries before the Europeans whose names they carry. And you will meet patterns that look perfect and then break, which is exactly why proof matters.

In earlier layers we wrote rules like “multiply the position by 3 and add 1”. Mathematicians shorten this with a variable, a letter that stands for any number. With n for the position, the rule becomes 3 × n + 1, often written even more briefly as 3n + 1 (a number written right in front of a letter means multiply).

A rule like 3n + 1 is an expression. When we say that the term T in position n is given by it, we write a formula: Tₙ = 3n + 1 (say “T sub n equals three n plus one”). The number multiplying n (here 3) is the coefficient; the number added on its own (here 1) is the constant.

This small step, using a letter for “any number”, is the beginning of algebra. It lets one line say something about infinitely many cases at once: Tₙ = 3n + 1 describes the 1st, the 10th, the 1,000th and the millionth picture of matchstick squares.

Here is the reason, once and for all, why an arithmetic sequence with step d always has a formula starting d × n.

Imagine the d times table: d, 2d, 3d, 4d, … Its nth term is d × n, and it goes up by d each time.

Your sequence also goes up by d each time. So your sequence and the d times table move in step: whatever the gap between them is at position 1, it stays exactly the same at every position. That gap is a − d (your first term minus the first term of the times table).

So Tₙ = (d times table) + (fixed gap) = d × n + (a − d).

For the matchstick squares, the 3 times table is 3, 6, 9, 12 and the sticks are 4, 7, 10, 13: always 1 more. So the rule is 3n + 1. For 50, 47, 44, 41, … (step −3), compare with −3, −6, −9, −12: the gap is always 53, so Tₙ = −3n + 53, better written 53 − 3n.

Different people see the same matchstick pattern in different ways, and each way gives a different-looking formula. Here are three ways to count n squares in a row:

How you see it	Counting	Formula
One stick, then 3 more per square	1 + 3 + 3 + …	1 + 3n
First square whole (4), then 3 per extra square	4 + 3 + 3 + …	4 + 3(n − 1)
Horizontal sticks and vertical sticks	n on top, n on the bottom, n + 1 upright	2n + (n + 1)
All squares separate, minus the shared sides	4 per square, minus n − 1 shared	4n − (n − 1)

All four formulas must give the same numbers, because they count the same sticks. And algebra agrees:

• 4 + 3(n − 1) = 4 + 3n − 3 = 3n + 1
• 2n + (n + 1) = 3n + 1
• 4n − (n − 1) = 4n − n + 1 = 3n + 1

Expressions that give the same value for every n are called equivalent expressions. Showing that two different pictures lead to equivalent expressions is a small proof that both ways of seeing are right.

A geometric sequence multiplies by the same number each time, so its terms are powers. We write 2 × 2 × 2 × 2 × 2 as 2⁵ (“2 to the power 5”); the small 5 is the index or exponent and counts how many 2s are multiplied.

• Powers of 2: 2⁰ = 1, 2¹ = 2, 2² = 4, … 2¹⁰ = 1,024 (about a thousand; that is why computers say “kilo” for 1,024 bytes).
• Powers of 10: 10⁰ = 1, 10¹ = 10, 10² = 100, 10³ = 1,000, … 10⁵ = 1,00,000 (one lakh), 10⁷ = 1,00,00,000 (one crore).
• Powers of 3: 1, 3, 9, 27, 81, 243, 729, …

For a geometric sequence starting at a with ratio r, the first term has had no multiplications, the second one, the third two, so the nth term is a × rⁿ⁻¹. That is why 3, 6, 12, 24, … has 10th term 3 × 2⁹ = 3 × 512 = 1,536.

Because 2¹⁰ ≈ 1,000, powers of 2 grow by roughly a factor of 1,000 every 10 steps: 2²⁰ ≈ 10 lakh (actually 1,048,576), and 2³⁰ ≈ 100 crore (actually 1,073,741,824).

For square numbers the first differences are not constant, but the second differences (the differences of the differences) are:

n	1	2	3	4	5	6
n²	1	4	9	16	25	36
1st difference		3	5	7	9	11
2nd difference			2	2	2	2

A constant first difference means a linear formula (like 3n + 1). A constant second difference means the formula has an n² in it: it is quadratic. For triangular numbers the second difference is 1, and indeed n(n + 1) ÷ 2 = (n² + n) ÷ 2 has n² in it. A constant ratio means an exponential (geometric) formula like 2ⁿ.

This gives you a checklist for any sequence: constant 1st difference → linear; constant 2nd difference → quadratic; constant ratio → exponential; each term the sum of the two before → Fibonacci-type.

Sanskrit and Prakrit poetry is built from syllables that are short (laghu, 1 beat) or long (guru, 2 beats). Poets and scholars of prosody (chandaḥśāstra, the science of metre) asked a natural question: how many different rhythms fill a line of exactly n beats?

Try it with S for a short syllable (1 beat) and L for a long one (2 beats):

Beats	Rhythms	How many
1	S	1
2	SS, L	2
3	SSS, SL, LS	3
4	SSSS, SSL, SLS, LSS, LL	5
5	SSSSS, SSSL, SSLS, SLSS, LSSS, SLL, LSL, LLS	8

The counts go 1, 2, 3, 5, 8, 13, 21, 34, …: the Fibonacci numbers! Why? Every rhythm of n beats ends in either a short syllable or a long one. If it ends in S, the rest is a rhythm of n − 1 beats; if it ends in L, the rest is a rhythm of n − 2 beats. So (rhythms of n) = (rhythms of n − 1) + (rhythms of n − 2). That is exactly the Fibonacci rule, and it is a real proof, not just a spotted pattern.

The ratios swing above and below a special number, getting closer each time: 1.6180339…, called the golden ratio and written φ (the Greek letter phi). It equals (1 + √5) ÷ 2.

This is the usual explanation for why Fibonacci numbers show up in plants. A growing sunflower adds each new seed at roughly a fixed angle of turn from the last, about 137.5°, which is 360° ÷ φ² (the “golden angle”). That angle packs the seeds evenly with no wasted gaps, and the spirals your eye picks out then come in neighbouring Fibonacci numbers: often 34 and 55, or 55 and 89 in large heads. Pinecones commonly show 8 and 13 spirals and pineapples 8, 13 and 21.

It is a strong tendency, not a law. In a citizen-science count of 657 sunflower heads, 565 of 768 careful spiral counts were Fibonacci numbers, but 136 of them — nearly one in five — had no Fibonacci structure at all. Some plants show neighbouring Lucas numbers (like 4, 7, 11) instead. So: common, and for a reason, but always worth counting rather than assuming.

Write a 1 at the top. Each row below starts and ends with 1, and every other number is the sum of the two numbers just above it. Halāyudha, explaining Pingala’s rules in the 10th century, called this the Meru Prastara, the “staircase of Mount Meru”. In Europe it is called Pascal’s triangle.

Pingala’s question was: in a line of n syllables, in how many ways can exactly k of them be long? The answer is the kth number in row n of the triangle (counting from 0). For example, in 4 syllables there are 6 ways to have exactly 2 long ones: LLSS, LSLS, LSSL, SLLS, SLSL, SSLL.

Why do the digits of multiples of 9 add to 9? Take a two-digit number with tens digit a and units digit b. Its value is 10a + b = 9a + (a + b). So the number and its digit sum a + b differ by 9a, a multiple of 9. If the number is a multiple of 9, its digit sum must be a multiple of 9 too. For 9 × 1 to 9 × 10 the digit sum is exactly 9. (For bigger multiples, like 99 = 9 × 11, the digit sum is 18, and adding those digits gives 9.) This is the reason behind the divisibility test for 9.

Why does 111 × 111 = 12,321? Multiply the long way. Each row of the working is 111, shifted one place:

			1	1	1
		1	1	1
	1	1	1
Column sums	1	2	3	2	1

The number of 1s in each column rises and falls, giving 1, 2, 3, 2, 1. With nine 1s the middle column holds 9, the biggest single digit. With ten 1s the middle column would hold 10, which forces a carry and the pattern breaks. You predicted this in Understand; now you know why.

Here is the most famous trap in the study of patterns. Draw a circle and mark some points on it. Join every pair of points with a straight line, and count the regions (pieces) the circle is cut into. Place the points so that no three lines meet at the same point inside, which gives the most regions.

Points	1	2	3	4	5	6
Regions	1	2	4	8	16	31

1, 2, 4, 8, 16, … Surely the next is 32? It is 31. Draw it very carefully with 6 points and count: 31 is the most you can get, however hard you try. (Watch out when you draw: six equally spaced points give only 30, because the three long diagonals all cross at the centre. Nudge one point and you get 31.) With 7 points you get 57, not 64, and with 10 points just 256, not 512.

The true rule has nothing to do with doubling. The regions are 1 + (number of lines) + (number of crossing points inside): each line and each crossing adds one region. With n points there are n(n − 1) ÷ 2 lines, and every choice of 4 points gives exactly one crossing. The doubling was a coincidence of small numbers.