Number Play
Why This Matters
Numbers are not just for adding and subtracting. You can play with them.
Think of a magic trick where a friend picks a number and you guess it. Or a Sudoku puzzle in the newspaper. Or a riddle where letters stand for hidden digits. All of these are number games. And every good number game hides a clever bit of maths inside.
This chapter is full of such games. A line of children calling out numbers. Squares where every row adds to the same total. A famous sequence that shows up in flowers and poems. Secret-code sums where you crack the code.
But here is the real point. You will not just play these games. You will learn the reason each one works. Once you know the reason, you can solve any puzzle of that kind, not just the one in front of you. That is the difference between guessing and knowing.
The Big Idea
A puzzle is fun, but the maths behind it is what makes you powerful. In this chapter, instead of trying every possibility by brute force, you learn to reason. Is a thing even possible at all? You can often answer that with a simple rule — like “odd numbers can never be split into equal pairs” — without checking a single case. Spotting the hidden rule turns a hard puzzle into an easy one. That is number play: looking for the reason, not just the answer.
Let’s Break It Down
Numbers can describe an arrangement
Here is a neat idea. Sometimes a number does not tell you “how much”. It tells you about a position or an arrangement.
Picture seven children standing in a line, all of different heights. Each child looks only at the children in front of them (to their left) and calls out one number: how many of those children are taller than them.
Let us see what that looks like.
The figure below shows one such line. Read each child’s number from left to right.
Look at Figure 6.1. Two facts pop out at once. First, the first child always says 0 — there is nobody in front of them to be taller. Second, the tallest child in the whole line always says 0 too, because no one can be taller than the tallest.
These give us a way to test statements as “always true”, “sometimes true”, or “never true”.
We just saw that the tallest person must say 0. So “if a person is the tallest, then their number is 0” is always true. But the other way round is different — saying 0 only means nobody in front is taller, not that you are the tallest of everyone. So “if a person says 0, they are the tallest” is only sometimes true.
Let us work out the biggest number anyone can ever say.
In a line of 8 people, what is the largest number any one person can call out?
- A person’s number counts how many taller people stand in front of them. To make this number as big as possible, we want as many people as possible in front, and all of them taller.
- The most people you can have in front of you is when you are the last person in the line. With 8 people, that means 7 people are in front of you.
- If you are the shortest of all 8, then every one of those 7 people in front is taller than you. So you count all 7.
- So the largest number possible in a group of 8 people is 7. (In general, in a group of n people the largest possible number is n − 1.)
Why can the first person in the line never say a number bigger than 0?
Because the number counts taller people standing in front. The first person has nobody in front of them at all, so there is nobody to count — their number must be 0.
Parity: the even–odd superpower
Here is one of the most useful ideas in all of maths, and it is very simple.
Every whole number is either even or odd. We call this its parity. Even numbers (2, 4, 6, 8, …) can be split into equal pairs with nothing left over. Odd numbers (1, 3, 5, 7, …) always have exactly one dot left over after pairing.
The picture below shows this clearly.
Figure 6.2 shows why the even–odd rules work — you can literally see it. Each odd number carries one leftover dot. Add two odds and the two leftovers pair up, so the sum is even. That is a real proof, made of dots.
From this single idea, all the rules follow:
even + even = even
odd + odd = even
even + odd = odd
Let us use this to solve a puzzle without checking every case.
Kishor has only odd-numbered cards. He wants to put one card in each of 5 boxes so the 5 cards add up to 30. Is it possible?
- All the cards are odd. He is adding 5 of them. So we are adding an odd number of odd numbers.
- Pair them up: odd + odd = even. So the first two cards give an even number. The next two give another even number. Even + even = even.
- That uses 4 cards and gives an even total. One odd card is still left. Even + odd = odd. So the sum of all 5 odd cards is always odd.
- But 30 is even. An odd total can never equal an even number. So no matter which cards he picks, it is impossible. He doesn’t need to try a single combination.
The same trick cracks an age riddle. Two siblings are born exactly one year apart, so their ages are consecutive numbers (like 11 and 12). In any two consecutive numbers, one is even and one is odd. Even + odd = odd. So their ages always add to an odd number. If someone claims the sum is 112 (even), that is impossible.
Parity even tells you about products (multiplication). Think of a 3 × 4 grid of small squares. It has 12 squares — even. A 3 × 3 grid has 9 — odd. The rule: a product is odd only when both numbers are odd. If even one of them is even, the product is even (because an even number of anything pairs up). So 27 × 13 is odd (both odd), but 42 × 78 is even (both even), and 135 × 654 is even (one is even).
Without multiplying, is 135 × 654 even or odd? Why?
Even. A product is odd only when both factors are odd. Here 654 is even, so the whole product is even — you don’t need to do the multiplication.
Sum grids: when a puzzle is impossible
Now a grid puzzle. Fill a 3 × 3 grid with the numbers 1 to 9, using each number exactly once. Outside the grid, circles show the sum of each row and each column.
The figure below shows a filled grid with its sums.
Look closely at Figure 6.3. The three row sums add to 16 + 9 + 20 = 45. The three column sums add to 13 + 17 + 15 = 45 too. This is no accident.
Here is the reason. Add all three row sums together and you have simply added every number in the grid once. The grid holds 1 to 9, and 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45. The columns cover the same nine numbers, so the column sums add to 45 as well.
This gives a quick “impossible” test. If a puzzle’s circle sums add to anything other than 45, it cannot be filled with 1 to 9. Also, the smallest possible row sum is 1 + 2 + 3 = 6, and the largest is 7 + 8 + 9 = 24. So no circle can hold a number below 6 or above 24. A puzzle asking for a row sum of 5 or 26 is impossible before you even start.
Magic squares
A magic square is a special filled grid where every row, every column, and both diagonals add up to the same total. That total is called the magic sum.
Here is a real 3 × 3 magic square made from 1 to 9.
Figure 6.4 shows all the lines working: rows, columns, and both diagonals all give 15. But why 15, and why is 5 in the centre? We can reason it out instead of guessing.
We just saw the three row sums add to 45. In a magic square all three are equal. Three equal numbers that add to 45 must each be 45 ÷ 3 = 15. So the magic sum has to be 15 — it can’t be anything else.
Now why must 5 sit in the centre? Let us check the corners.
Show that the centre of a 3 × 3 magic square (using 1 to 9) must be 5.
- The centre number is part of four lines at once: its row, its column, and both diagonals. So it gets used a lot.
- Try a big number like 9 in the centre. Every line through it must total 15, so the other two numbers on each line must add to 15 − 9 = 6. But you’d need several different pairs that each add to 6 using 1 to 8, and there just aren’t enough. It fails.
- Try a small number like 1 in the centre. Each line needs the other two to add to 15 − 1 = 14, again needing many different pairs. There aren’t enough. It fails too.
- Working through the numbers this way, only 5 leaves enough matching pairs (1+9, 2+8, 3+7, 4+6 all give 10, and 5 + 10 = 15). So the centre must be 5.
There is more lovely history here. The oldest known magic square, the Lo Shu square, comes from ancient China over 2000 years ago. The first known 4 × 4 magic square is carved into a 10th-century temple at Khajuraho in India — it is called the Chautisa Yantra because every line adds to 34 (chautis means 34 in Hindi). Indian mathematicians wrote down general methods to build them. So this “game” is part of real mathematical history.
In a 3 × 3 magic square made from 1 to 9, why does the magic sum have to be exactly 15?
The three row sums add to 45 (the total of 1 to 9). In a magic square all three rows are equal, so each must be 45 ÷ 3 = 15. There is no other choice.
The Virahanka numbers (nature’s favourite sequence)
Now meet one of the most famous sequences in all of mathematics:
1, 2, 3, 5, 8, 13, 21, 34, 55, …
The rule is beautiful and simple: each number is the sum of the two before it. 1 + 2 = 3. 2 + 3 = 5. 3 + 5 = 8. 5 + 8 = 13. And on it goes.
The picture below shows the rule in action.
Look at the bottom of Figure 6.5. This sequence was first discovered in India, in the study of poetry, around 700 CE by the Prakrit scholar Virahanka — about 500 years before Fibonacci wrote about it in Italy. That is why we call them the Virahanka–Fibonacci numbers.
How did poetry give us numbers? In many Indian languages each syllable is either short (1 beat) or long (2 beats). Ask: in how many ways can you fill a line of a given number of beats? That is the same as asking: in how many ways can you write a number as a sum of 1s and 2s?
Let us count for 3 beats.
In how many ways can you write 3 as a sum of 1s and 2s (counting different orders separately)?
- List them carefully. Start with all 1s: 1 + 1 + 1. That’s one way.
- Use a 2 at the start: 2 + 1. That’s a second way.
- Use a 2 at the end: 1 + 2. That’s a third way. (2 + 1 and 1 + 2 count as different rhythms.)
- There are no more. So there are 3 ways. This matches the 3 in the sequence 1, 2, 3, …
Now the clever part — why each term is the sum of the two before. Every rhythm starts with either a “1+” or a “2+”. If it starts with “1+”, the rest must fill (beats − 1) beats. If it starts with “2+”, the rest must fill (beats − 2) beats. So the number of rhythms for n beats equals the rhythms for (n−1) beats plus the rhythms for (n−2) beats. That is exactly the add-the-two-before rule!
This sequence also hides in nature. The number of petals on a daisy is very often a Virahanka number — 13, 21, or 34.
Two terms of the Virahanka sequence are 21 and 34. What is the next term, and how do you know?
- Each term is the sum of the two before it, and 21 + 34 = 55.
Cryptarithms: digits in disguise
The last game is a code-cracker. In a cryptarithm, digits are hidden behind letters. Each letter stands for one digit (0 to 9), and the same letter always means the same digit. Your job is to figure out which digit each letter is.
Here is a classic one: T + T + T = UT (a two-digit answer).
The figure below walks through cracking it.
The reasoning in Figure 6.6 is the whole skill. Don’t guess wildly — find the clue. Here, three copies of T must add to a number whose last digit is T again. That is a strong clue. Let us pin it down.
Crack the cryptarithm T + T + T = UT, where U and T are single digits.
- T + T + T is just 3 × T. The answer UT is a two-digit number whose units (last) digit is T. So 3 × T must end in T.
- Test the digits. 3 × 1 = 3 (ends in 3, not 1 — no). 3 × 2 = 6 (no). 3 × 3 = 9 (no). 3 × 4 = 12 (ends in 2 — no). 3 × 5 = 15 (ends in 5 — yes!).
- So T = 5. Then UT = 3 × 5 = 15, which means U = 1 and T = 5. Check: it is a two-digit number ending in 5. Correct.
- The answer: T = 5, U = 1, so the sum reads 5 + 5 + 5 = 15.
Parity helps with cryptarithms too. A “toggle” puzzle is a parity puzzle in disguise: a switch flipped an even number of times returns to its start, and flipped an odd number of times ends in the opposite state. So a bulb that is ON and toggled 77 times (odd) ends OFF.
Common Mistakes
Watch out for these traps. Each one feels right at first.
If a person in the height line says '0', they must be the tallest in the whole group.
The tallest person really does say 0, so it is tempting to flip the statement and assume anyone who says 0 must be the tallest.
Saying 0 only means nobody standing IN FRONT is taller. A short person at the very front of the line also says 0, because there is no one ahead of them. So '0' means tallest-so-far, not tallest overall — it is only sometimes the tallest.
Adding five odd numbers could give an even total if you pick the right ones.
With even numbers any total feels reachable, so it seems like odd numbers should be just as flexible if you hunt for the right combination.
Pair the odds two at a time: odd + odd = even. Four odds give an even total, and the fifth odd makes it odd again. So an odd count of odd numbers is ALWAYS odd — no combination can make it even. 30 is impossible.
In a magic square you just need every row and every column to add to the same number.
Rows and columns are the obvious lines you check first, so it feels like getting those right is the whole job.
A true magic square also needs both DIAGONALS to add to the same magic sum. A grid with equal rows and columns but unequal diagonals is not a magic square.
In the sequence 1, 2, 3, 5, 8, … the rule is 'add 1, then add 1, then add 2…' or some changing step.
The early gaps (1 to 2, 2 to 3) are both just +1, so it looks like the steps follow their own little pattern.
The real rule is fixed: each term is the SUM of the two terms before it. 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13. The growing gaps are a result of that one rule, not a separate pattern.
Quick Check
Test yourself. Pick an answer, then read why.
A ₹1 coin and a ₹5 coin are both odd in value. Lakpa has an ODD number of ₹1 coins, an ODD number of ₹5 coins and an EVEN number of ₹10 coins. He says his total is ₹205. Did he make a mistake?
In a 3 × 3 magic square made from the numbers 1 to 9, what is the magic sum?
The Virahanka sequence is 1, 2, 3, 5, 8, 13, 21, … What number comes right after 21?
Practice Problems
Try each one yourself first. Then reveal the full solution.
Easy
A light bulb is ON. Dorjee toggles its switch 77 times. Will the bulb end up ON or OFF?
Each toggle flips the bulb to the opposite state. Two toggles bring it back to where it started. So what matters is the parity of the number of toggles.
77 is odd. An odd number of flips leaves the bulb in the opposite state to where it began. It started ON, so it ends OFF.
Find the parity (even or odd) of the sum: 2 odd numbers and 3 even numbers added together.
Take the odd numbers first: odd + odd = even. The even numbers add up to an even number no matter how many there are. So we have even + even = even.
The total is even.
Medium
Liswini's encyclopaedia loses 50 loose sheets, each printed on both sides. Can the sum of all the page numbers on these sheets be 6000? Why or why not?
Each sheet has two sides, so two page numbers. The two sides of one sheet are always a front and a back, which are consecutive numbers, like 11 and 12. One is even and one is odd, so the two pages on a sheet add to an odd number.
There are 50 sheets, each contributing an odd sum. So we are adding 50 odd numbers together. 50 is even, and an even count of odd numbers gives an even total.
6000 is even, so the parity matches — it is not impossible on parity grounds. (Whether an exact set of 50 sheets sums to precisely 6000 depends on the book, but parity alone does not rule it out.)
Two consecutive Virahanka numbers are 987 and 1597. Find the next two numbers, and the previous two numbers.
Going forward, each term is the sum of the two before it. Next term = 987 + 1597 = 2584. Term after that = 1597 + 2584 = 4181.
Going backward, the term before 987 is the number that, added to 987, gives 1597. So it is 1597 − 987 = 610. The term before that is 987 − 610 = 377.
So the sequence around them is: …, 377, 610, 987, 1597, 2584, 4181, …
Challenge
Angaan climbs an 8-step staircase. He can take either 1 step or 2 steps at a time. In how many different ways can he reach the top?
This is exactly the Virahanka rhythm puzzle in disguise. Each “path” is a way of writing 8 as a sum of 1s and 2s, and order matters.
Let W(n) be the number of ways to climb n steps. Every path starts with either a 1-step (leaving n − 1 steps) or a 2-step (leaving n − 2 steps). So W(n) = W(n−1) + W(n−2) — the add-the-two-before rule.
Start small and build up:
- W(1) = 1 (just one 1-step)
- W(2) = 2 (1+1, or 2)
- W(3) = 1 + 2 = 3
- W(4) = 2 + 3 = 5
- W(5) = 3 + 5 = 8
- W(6) = 5 + 8 = 13
- W(7) = 8 + 13 = 21
- W(8) = 13 + 21 = 34
So there are 34 different ways to climb the 8-step staircase.
Solve the cryptarithm: K2 + K2 = HMM. (K2 means a two-digit number with K in the tens place and 2 in the units place. HMM is a three-digit answer.)
We are adding a two-digit number to itself: K2 + K2 = 2 × K2. The answer HMM has three digits, so 2 × K2 is at least 100. That needs K to be large.
Look at the units column first: 2 + 2 = 4, with no carry. So the units digit M = 4.
Now the tens digit of the answer is also M = 4. The tens column is K + K = 2K (plus any carry from units — but there was none). The result must end in 4 and carry 1 into the hundreds (to make H = 1, since doubling a two-digit number gives at most a 1 in the hundreds place).
So 2K must give a tens-digit of 4 with a carry of 1 — that means 2K = 14, so K = 7.
Check: K2 = 72, and 72 + 72 = 144. So H = 1, M = 4, K = 7. HMM = 144. Correct.
Summary
- A number can describe an arrangement, like each child saying how many taller people stand in front. The first person, and the tallest person, always say 0.
- Parity is whether a number is even or odd. Even numbers split into equal pairs; odd numbers leave exactly one over.
- The parity rules: even + even = even, odd + odd = even, even + odd = odd. A product is odd only when both numbers are odd.
- Parity lets you prove something is impossible without checking cases — like 5 odd cards never summing to 30.
- In a sum grid, all the row sums (and all the column sums) of a 1-to-9 grid add to 45, because that is the total of 1 + 2 + … + 9.
- A magic square has every row, column, and both diagonals equal. For 1 to 9 the magic sum must be 15 and the centre must be 5 — and you can reason out why.
- The Virahanka–Fibonacci sequence 1, 2, 3, 5, 8, 13, 21, 34, … adds the two terms before to get the next. It was discovered in India through poetry, around 700 CE.
- A cryptarithm hides digits behind letters. You crack it by finding clues (like a units digit that must repeat) instead of guessing.
What’s Next
You have spent this chapter playing with numbers and finding the hidden reasons behind each game. Next, you turn from numbers to shapes.
In Chapter 7 — A Tale of Three Intersecting Lines, you will explore triangles: how three lines meet to make one, what rules their sides and angles must follow, and why some triangles can be built while others simply cannot. The same habit you built here — asking “is this even possible, and why?” — will be your best tool there too.
Frequently Asked Questions
What is parity in maths and how is it used to solve puzzles?
Parity means whether a number is odd or even. An even number can be split into equal pairs; an odd number always has one left over. In puzzles, you can often prove something is impossible simply by checking parity — for example, you can never split an odd total into two equal whole-number groups.
How does a 3x3 magic square work and what is special about it?
In a 3x3 magic square, nine different numbers are placed in a 3x3 grid so that every row, every column, and both diagonals add up to the same total (the 'magic sum'). The classic square using 1 to 9 has a magic sum of 15. The centre cell always holds the middle value of the set.
What are Virahanka numbers and how are they formed?
The Virahanka sequence (also known as Fibonacci numbers) starts with 1, 1, and then every number after that is the sum of the two numbers before it: 1, 1, 2, 3, 5, 8, 13, 21, and so on. They were first described by the Indian poet-mathematician Virahanka to count rhythm patterns in Sanskrit poetry.
What is a cryptarithm puzzle and how do you solve it?
A cryptarithm is an addition or multiplication puzzle where letters replace the digits, and you have to figure out which digit each letter stands for. You solve it by using logic: look at carries, check what digits are possible for each column, and eliminate options until only one solution fits.
Why can some number arrangements be ruled out without trying every possibility?
Because of parity rules and fixed totals, you can often show an arrangement is impossible in one step. For example, if a puzzle needs an odd number of things arranged in equal groups, you know immediately it cannot work — no need to try. Spotting the rule saves a lot of time compared to brute-force guessing.