Number Play

Take any whole number and ask one simple question: can it be split into pairs with nothing left over? If the answer is yes, the number is even — like 24, which splits into 12 neat pairs. If one item is always left over, the number is odd — like 25, which gives 12 pairs and one lonely leftover. This even-or-odd property is called a number's parity, and it is a surprisingly powerful shortcut for predicting an answer before you've even finished calculating.

Addition has a clean parity pattern. Add two even numbers and you always get an even number — two complete sets of pairs simply combine into a bigger set of pairs. Add two odd numbers and the two leftover items pair up with each other, so the sum is even again: 7 + 9 = 16. But add an even number and an odd number, and that one leftover item has nobody to pair with, so the sum is odd: 12 + 7 = 19.

Multiplication follows its own pattern, and it's easy to get wrong if you guess instead of reasoning it through. Even × even is even (4 × 6 = 24). Odd × odd is odd, because no factor of 2 ever enters the product (3 × 5 = 15). But even × odd is even, and odd × even is even too (4 × 3 = 12, and 3 × 4 = 12) — because the single even factor already guarantees the product can be split into pairs, whatever it's multiplied by. In short: a product is odd only when every number being multiplied is odd. If even one factor is even, the whole product is even.

This reasoning extends to algebraic expressions too. An expression like 2n + 1 is always odd, whatever whole number n you substitute, because 2n is always even (it carries a guaranteed factor of 2) and adding 1 to an even number always breaks exactly one pair. Expressions like 2n or 100p are always even for the same reason — they are built from a factor of 2.

Long division always works, but it's slow when all you want to know is whether a number divides evenly. For 2, only the last digit matters: if it's 0, 2, 4, 6 or 8, the number is even. For 5, only the last digit matters again: it must be 0 or 5. For 10, the last digit must be 0 — which makes sense, since 10 = 2 × 5, so the number needs both of those rules satisfied at once.

Some rules look at the last two or three digits instead of just one. A number is divisible by 4 exactly when its last two digits, read as their own two-digit number, are divisible by 4 — for 61,600, the last two digits are "00", which is divisible by 4, so the whole number is too. A number is divisible by 8 exactly when its last three digits are divisible by 8 — for 83,160, the last three digits "160" give 160 ÷ 8 = 20 exactly, so 83,160 is divisible by 8.

Other rules look at the digit sum. Add up every digit of a number: if that sum is divisible by 3, so is the original number. For 4,536, checking the last-three-digits rule for 8 is more useful: "536" gives 536 ÷ 8 = 67 exactly, so 4,536 is divisible by 8. For 9, the rule is the same idea but stricter — the digit sum itself must be divisible by 9. Take 7,938: its digits add to 7 + 9 + 3 + 8 = 27, and 27 is divisible by 9, so 7,938 is divisible by 9 (indeed, 7,938 ÷ 9 = 882 exactly). Since 6 = 2 × 3, a number is divisible by 6 only when it passes the rule for 2 and the rule for 3 at once — 9,042 is even, and its digits add to 15 (divisible by 3), so it passes both and is divisible by 6.

The rule for 11 is the most surprising one: starting from the rightmost digit, alternately add and subtract the digits. If the result is divisible by 11 (including 0), so is the number. For 2,728, working right to left gives 8 − 2 + 7 − 2 = 11, which is divisible by 11 — and indeed 2,728 ÷ 11 = 248 exactly. Try the same check on 1,234: 4 − 3 + 2 − 1 = 2, which is not divisible by 11, so 1,234 is not either.

A magic square is a grid of numbers where every row, every column and both diagonals add up to exactly the same number, called the magic sum. The classic version uses the numbers 1 to 9 in a 3×3 grid. Before guessing where each number should go, you can reason out the magic sum itself: adding every number from 1 to 9 gives 1+2+3+4+5+6+7+8+9 = 45, and since the 3 rows share this total equally, each row — and so the magic sum — must be 45 ÷ 3 = 15.

A similar trick reveals which number must sit at the very centre. The centre cell belongs to four different lines — its row, its column and both diagonals — and all four of those lines add up to 15, giving a combined total of 4 × 15 = 60. In that combined total, the centre number gets counted four times, while each of the other eight numbers gets counted exactly once (each one pairs with its "opposite" number through the centre). So 4 × (centre) + (sum of the other eight numbers) = 60. Since all nine numbers 1 to 9 add to 45, the other eight numbers add to 45 − (centre). Substituting: 4 × (centre) + 45 − (centre) = 60, which simplifies to 3 × (centre) = 15, so the centre must be 5 — always, however the rest of the square is arranged.

Magic squares have a long history in India. The oldest known 4×4 magic square in the world is carved into a 10th-century inscription at a Jain temple in Khajuraho — it is called the Chautisa Yantra ("Chautisa" means 34), because every row, column and diagonal in it adds up to 34. That number checks out with the same reasoning used above: the numbers 1 to 16 add up to 136, and sharing that across 4 rows gives 136 ÷ 4 = 34. Magic squares carved into temple pillars have also been found in Tamil Nadu, and designs like the Navagraha Yantra and Kubera Yantra are still used decoratively in Indian homes and shops today.

Once you have one working magic square, you can transform it into new ones. Adding the same number to every cell keeps it magic (just with a new magic sum), and doubling every cell also keeps it magic — because changing every number in a row by the same operation changes every row's sum by the same amount, so if all the rows were equal before, they stay equal afterwards.

In the poetry of many Indian languages — Prakrit, Sanskrit, Marathi, Malayalam and more — every syllable is either short (one beat) or long (exactly two beats). Ancient poets and linguists asked a very mathematical question: in how many different ways can you fill a rhythm of a given number of beats using only short (1-beat) and long (2-beat) syllables? For 1 beat, there is only 1 way. For 2 beats, there are 2 ways (two shorts, or one long). For 3 beats, there are 3 ways. For 4 beats, there are 5 ways. The pattern that emerges — 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... — is built by a beautifully simple rule: every number is the sum of the two numbers before it (3 = 1+2, 5 = 2+3, 8 = 3+5, and so on).

This sequence was first explicitly written down around 700 CE by the Prakrit scholar Virahanka, who explained the counting rule in the form of a Prakrit poem. Virahanka built on earlier ideas from the Sanskrit scholar Pingala (around 300 BCE), and the sequence was studied further by Gopala (around 1135 CE) and Hemachandra (around 1150 CE) — all well before it reached Europe. The Italian mathematician Fibonacci wrote about the very same numbers in 1202 CE, roughly five hundred years after Virahanka first described them. Because of this history, the sequence is often called the Virahanka–Fibonacci sequence, crediting its true origin.

The sequence has a striking parity pattern too. Writing the terms as Odd or Even — 1(O), 2(E), 3(O), 5(O), 8(E), 13(O), 21(O), 34(E), 55(O), 89(O), 144(E) — shows a repeating block of Odd, Even, Odd, Odd, Even that settles into every third term (the 2nd, 5th, 8th, 11th, and so on) being even, with the rest odd. This happens because odd + even = odd (continuing the pattern) while odd + odd = even (resetting it) — the same parity rules from earlier in this chapter, now explaining a famous sequence.

The Virahanka–Fibonacci numbers show up constantly outside poetry and mathematics — the number of petals on many flowers (13, 21 or 34, for instance) and the spiral patterns on pinecones and sunflower heads often match numbers from this very sequence, remarkable given that it was first discovered by people counting syllables in verse.

A cryptarithm (also called an alphametic) replaces the digits in a sum with letters, where every occurrence of the same letter always stands for the same digit (0-9), and different letters stand for different digits. Your job is to work out which digit each letter represents, using nothing but logical reasoning about how addition works — not guesswork.

Consider the puzzle T + T + T = UT, where T and U are single digits and UT is a two-digit number with U in the tens place and T in the units place. Since T is added to itself three times, the units digit of 3 × T must come out to T itself. Testing digits systematically: if T = 5, then T + T + T = 15, and the units digit of 15 is indeed 5, matching T, with U = 1 in the tens place. No other single digit works this cleanly, so T = 5 and U = 1, giving 5 + 5 + 5 = 15.

A trickier example is K2 + K2 = HMM, where K2 means a two-digit number with units digit 2 and tens digit K, and HMM is a three-digit number whose tens and units digits are both M. Since 2 + 2 = 4, the units digit of the sum must be 4, so M = 4. Now you need K2 + K2 to actually equal a number ending in "44", with a nonzero hundreds digit H. Testing K = 7 gives K2 = 72, so 72 + 72 = 144 — and that result does end in "44", with H = 1. So K = 7, M = 4 and H = 1.

The general strategy for any cryptarithm is the same: start from the units column, since it is affected by the fewest unknowns, use whatever digit you discover there to work out the next column, and keep carrying that information forward — exactly like a detective using one solid clue to unlock the next.