Pattern hunters: puzzles, projects and open questions

You now know how to spot, describe, predict and prove patterns. This last layer takes you out into the wide world of patterns: squares of numbers that are magic in every direction (one carved on a temple wall at Khajuraho in medieval India), floors and rangoli that tile the plane, number shapes beyond squares and triangles, patterns that go round in cycles, a legend about rice on a chessboard, olympiad-style puzzles, patterns in music, cricket and calendars, projects to try, and questions that nobody in the world has answered yet.

Dip in anywhere. Each chapter stands on its own.

A magic square is a square grid of numbers in which every row, every column and both main diagonals add up to the same total, called the magic constant. Usually the numbers are 1, 2, 3, … up to the number of cells, each used once.

The oldest known is the Chinese Lo Shu square, 3 by 3, said in legend to have appeared on the back of a turtle:

4	9	2
3	5	7
8	1	6

Every line adds to 15: 4 + 9 + 2, 3 + 5 + 7, 4 + 5 + 6, and so on.

Why 15? The numbers 1 to 9 add up to 45. The three rows share out all nine numbers, so each row must be 45 ÷ 3 = 15. In general, a square of side n using 1 to n² has magic constant n(n² + 1) ÷ 2: 15 for 3 × 3, 34 for 4 × 4, 65 for 5 × 5, 260 for 8 × 8.

In India, magic squares were studied in depth. Nārāyaṇa Paṇḍita’s Gaṇita Kaumudī (1356) gave general methods for building them of many sizes, and they appear in older texts too, sometimes linked to ideas of good luck or medicine.

At the Parshvanath Jain temple in Khajuraho (Madhya Pradesh) a 4 by 4 magic square is carved into the stone:

7	12	1	14
2	13	8	11
16	3	10	5
9	6	15	4

How old is it? The temple itself was built about 950–970 CE, but the inscription is usually dated a little later, to around the 12th century. You will often see “10th century” quoted, from the date of the temple rather than of the carving.

It uses 1 to 16 once each, and every row, column and main diagonal adds to 34 — which is why it is known as the Chautisa Yantra (chautisa is Hindi for thirty-four). But it does far more than that:

• The broken diagonals also add to 34. For example 12 + 8 + 5 + 9 (wrapping round the edge) = 34. A square like this is called pandiagonal.
• Every 2 by 2 block of four neighbouring cells adds to 34, even blocks that wrap round the edges: 7 + 12 + 2 + 13 = 34, 10 + 5 + 15 + 4 = 34.
• Numbers two steps apart along a diagonal add to 17: 7 + 10, 12 + 5, 13 + 4, …

Squares with all these properties are called most-perfect magic squares. The Khajuraho square is one of the oldest surviving examples anywhere in the world, and studying it is what taught European mathematicians in the late 19th century to care about pandiagonal squares.

A tessellation (or tiling) is a pattern of shapes that covers a flat surface with no gaps and no overlaps, and can go on for ever. Floor tiles, brick walls, honeycombs, jali screens, the grid of dots under a kolam, and the repeating designs on block-printed cloth are all tessellations.

Which regular polygons (all sides and angles equal) can tile a floor on their own? Where the corners meet, the angles must add up to exactly 360°:

Shape	Each angle	How many fit round a point?
Equilateral triangle	60°	360 ÷ 60 = 6 ✓
Square	90°	360 ÷ 90 = 4 ✓
Regular pentagon	108°	360 ÷ 108 = 3.33… ✗
Regular hexagon	120°	360 ÷ 120 = 3 ✓
Regular octagon	135°	360 ÷ 135 = 2.67 ✗

Only three regular polygons tile on their own: triangles, squares and hexagons. That is why bees use hexagons: of the three, hexagons enclose the most space for the least wall.

Mix shapes and more is possible. Octagons and squares together (135° + 135° + 90° = 360°) make a common bathroom-floor pattern. Triangles and hexagons (60° + 60° + 120° + 120°) make a pattern found in old Indian and Islamic jali work.

Rangoli, kolam and many textile patterns combine tessellation with symmetry:

• Reflection symmetry: one half is a mirror image of the other.
• Rotational symmetry: the design looks the same after a turn. A rangoli flower with 8 petals looks the same after 1/8 of a turn (45°), so it has rotational symmetry of order 8.
• Translation symmetry: a border pattern looks the same after sliding along by one unit. This is the repeating unit you met in Discover.

Kolam traditions have their own deep mathematics: some kolam are drawn as a single unbroken line that loops around every dot and returns to its start. Computer scientists have studied these as examples of patterns generated by simple rules, and designs like them are now used to teach coding.

Numbers that can be arranged as dots in a regular shape are called figurate numbers. The Greeks loved them, and they appear all over puzzles.

• Centred hexagonal numbers 1, 7, 19, 37, 61, … are one dot surrounded by rings of 6, 12, 18, 24 dots. Look at a pack of pencils or a bundle of straws held together: 7 and 19 are the natural sizes. Their running totals are the cubes: 1 = 1, 1 + 7 = 8, 1 + 7 + 19 = 27!
• Tetrahedral numbers 1, 4, 10, 20, 35, … count oranges stacked as a triangular pyramid at a fruit stall.
• Square pyramidal numbers count balls stacked as a square pyramid. They also answer a classic puzzle: how many squares of all sizes are on a chessboard? There are 64 small ones, 49 of size 2 × 2, 36 of size 3 × 3, … down to 1 of size 8 × 8: 1 + 4 + 9 + … + 64 = 204.

Some growing patterns hide a repeating pattern inside them. The powers of 7 grow enormous: 7, 49, 343, 2,401, 16,807, … but their last digits go 7, 9, 3, 1, 7, 9, 3, 1, … a cycle of length 4.

Powers of	Last digits	Cycle length
2	2, 4, 8, 6, 2, 4, 8, 6, …	4
3	3, 9, 7, 1, 3, 9, 7, 1, …	4
4	4, 6, 4, 6, 4, 6, 4, 6, …	2
7	7, 9, 3, 1, 7, 9, 3, 1, …	4
9	9, 1, 9, 1, 9, 1, 9, 1, …	2
5	5, 5, 5, 5, 5, 5, 5, 5, …	1

This lets you answer questions that look impossible. What is the last digit of 7²⁰²⁶? The cycle has length 4, and 2026 ÷ 4 = 506 remainder 2, so 7²⁰²⁶ ends like 7², in 9. It is exactly the “which bead is 20th?” trick from Discover, applied to a number with 1,713 digits.

The calendar's great cycle. A year of 365 days is 52 weeks and 1 day, so the same date moves on one weekday each year (two after 29 February). Leap years follow a three-part rule: a year divisible by 4 is a leap year, except century years, which are leap years only if divisible by 400. So 2000 was a leap year, but 1900 was not and 2100 will not be.

In 400 years there are 97 leap years, giving 400 × 365 + 97 = 146,097 days, which is exactly 20,871 weeks. So the Gregorian calendar repeats perfectly every 400 years: 1 January 2426 will fall on the same weekday as 1 January 2026, a Thursday.

An old legend, told in India and Persia, says that the inventor of chess asked the king for a humble reward: one grain of rice on the first square of the chessboard, two on the second, four on the third, doubling on every square up to the 64th. The king laughed at such a small request, and then his treasurers started counting.

• Square 10: 2⁹ = 512 grains.
• Square 20: 2¹⁹ = 524,288 grains.
• Square 64: 2⁶³ = 9,223,372,036,854,775,808 grains.
• The whole board: 1 + 2 + 4 + … + 2⁶³ = 2⁶⁴ − 1 = 18,446,744,073,709,551,615 grains (using the doubling-sum trick from Deepen).

A grain of rice weighs roughly 0.025 g (a thousand grains weigh about 20–30 g). At 0.025 g a grain, the whole board comes to about 461 thousand million tonnes of rice. The world grows around 550 million tonnes of rice a year, so that is more than 800 years of the entire world’s rice harvest. The king could not possibly pay.

Doubling also gives us binary, the number system inside every computer. In our everyday decimal system each place is worth 10 times the place to its right (ones, tens, hundreds, thousands). In binary each place is worth 2 times the place to its right: ones, twos, fours, eights, sixteens, …, and the only digits are 0 and 1.

Decimal	Made from powers of 2	Binary
5	4 + 1	101
13	8 + 4 + 1	1101
20	16 + 4	10100
100	64 + 32 + 4	1100100
255	128 + 64 + 32 + 16 + 8 + 4 + 2 + 1	11111111

Counting in binary makes its own pattern: 1, 10, 11, 100, 101, 110, 111, 1000, … The last digit alternates 1, 0, 1, 0 (odd, even); the second-last goes 0, 1, 1, 0, 0, 1, 1, … in pairs; the next in fours. Pingala’s rules for listing patterns of short and long syllables, over 2,000 years ago, work just like counting in binary with S and L instead of 0 and 1.

Almost every career that uses mathematics is really about patterns: an actuary predicts future costs from past data, a cryptographer designs and breaks codes, an architect tiles floors and repeats structural units, a textile designer creates repeats that line up at the edges of fabric, a composer builds music from repeated and varied phrases, a biologist models how populations grow, and a software engineer writes loops and recursive functions all day. The habit you have practised in this topic, notice, describe, predict, check, explain, is useful in all of them.

Here is a pattern a ten-year-old can try and no mathematician in the world can prove. Start with any whole number. If it is even, halve it. If it is odd, multiply by 3 and add 1. Repeat.

• From 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1.
• From 7: 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.
• From 27: it takes 111 steps, climbing as high as 9,232 before finally falling to 1.

The Collatz conjecture (1937) says that every starting number eventually reaches 1. Computers have checked every starting number up to more than 10²⁰ and all of them reach 1. Yet nobody has a proof. One of the most famous mathematicians of the 20th century, Paul Erdős, said that mathematics “is not yet ready for such problems”.