Number Play

Why This Matters

You already know how to count. You can add, subtract, multiply and divide. That is a lot!

But numbers can do more than just count things. Numbers can also play. They hide patterns. They hide little surprises. And when you look closely, you start to see those patterns everywhere.

This chapter is all about having fun with numbers. We will look at a line of children and use a number to describe it. We will hunt for special boxes called “supercells”. We will make the biggest and smallest numbers we can from a few digits. We will meet numbers that read the same forwards and backwards. And we will see a magic trick where almost any number you start with leads to the same secret number: 6174.

None of this is hard. There is nothing to memorise. You just need to be curious and ask “why does this happen?” That question is the real magic. Let’s play.

The Big Idea

Numbers are not just for counting. They can describe things, they hide patterns, and they follow rules you can discover yourself. Once you look at numbers as a game — biggest, smallest, same-both-ways, supercell — you stop fearing them and start playing with them. And every fun trick in this chapter has a clear reason behind it. We will not just see that it works. We will always see why it works.

Let’s Break It Down

Numbers can describe a picture

A number does not always mean “how many apples”. Sometimes a number describes something about an arrangement.

Here is a fun game from the chapter. Some children stand in a line. They are all of different heights. Each child looks only at the children right next to them — their neighbours. Then each child says a number:

• They say 0 if no neighbour is taller than them.
• They say 1 if exactly one neighbour is taller.
• They say 2 if both neighbours are taller.

So the number each child says is just how many of their neighbours are taller.

A child at the end of the line has only one neighbour (there is nobody on one side). So an end child can only say 0 or 1, never 2.

Figure 3.1 below shows five children in a line and the number each one says.

Tap to zoom

Notice something nice. The tallest child of all will always say 0, because nobody is taller than them. And a child who says 2 must be shorter than both neighbours, so they are like a little dip in the line.

Let’s think carefully about one of the chapter’s puzzles.

That kind of reasoning — “think about the tallest one” — is exactly how we solve these number puzzles. We do not check every arrangement one by one. We find a reason.

Supercells — the biggest in the neighbourhood

Now look at a row of boxes (we call each box a cell). Each cell has a number in it.

A cell is called a supercell if its number is bigger than the numbers in the cells right next to it (its neighbours). A cell in the middle has two neighbours, one on each side. A cell at the end of the row has only one neighbour.

Figure 3.2 below shows a row of eight cells. The supercells are coloured yellow.

Tap to zoom

Let’s check one cell fully so the rule is clear.

Here is a “why” question that the chapter loves to ask. Think before you read the answer.

Making the biggest and smallest numbers

Suppose someone gives you a few digits, say 7, 4, 3 and 2, and asks you to arrange them into a 4-digit number. You could make many different numbers: 2347, 4732, 7234, and so on.

Which arrangement is the biggest? And which is the smallest?

Here is the key idea. In a number, the digit on the left is worth the most. The leftmost spot is the thousands place, the next is hundreds, then tens, then ones. A bigger digit in the thousands place adds a lot more value than the same digit in the ones place.

So to make the biggest number, put your biggest digit on the left, then the next biggest, and so on — large to small. To make the smallest number, do the opposite — small to large.

Figure 3.3 below shows both arrangements for the digits 7, 4, 3, 2.

Tap to zoom

Let’s do a full example.

That little 0 trap is easy to miss, so go slow whenever a 0 is one of your digits.

Playing with digits — digit sums and patterns

The digit sum of a number is just what you get when you add up all its digits.

For example, the digit sum of 68 is 6 + 8 = 14. The digit sum of 176 is 1 + 7 + 6 = 14 too. And the digit sum of 545 is 5 + 4 + 5 = 14. So 68, 176 and 545 all have the same digit sum, 14, even though they look very different.

Digit sums hide neat patterns. The chapter asks you to find the digit sums of numbers in a row, like 40 to 70, and watch what happens. Try 40, 41, 42, 43… Their digit sums are 4, 5, 6, 7… The digit sum goes up by 1 each time. Then at 50 it suddenly drops back to 5. Spotting these little jumps is the fun part.

Palindromes — same both ways

Some numbers read the same from left to right and from right to left. Try reading 575 backwards — you get 575 again! Numbers like this are called palindromes.

Other palindromes: 66, 848, 797, 1111. (The word “palindrome” is also used for words like MOM or LEVEL, which read the same both ways.)

Figure 3.4 below shows why 575 is a palindrome, and a fun game you can play with palindromes.

Tap to zoom

There is a fun game with palindromes called reverse and add. Start with any 2-digit number. Reverse its digits and add the two numbers. If the answer is a palindrome, stop. If not, repeat the steps.

The magic number 6174 (Kaprekar)

This is the most surprising trick in the whole chapter. It was found by an Indian maths teacher, D. R. Kaprekar, from Maharashtra, who simply loved playing with numbers.

Here are the steps. Take any 4-digit number that has at least two different digits (so not 1111 or 2222). Then:

1. Make the biggest number from its digits. Call it A.
2. Make the smallest number from its digits. Call it B.
3. Subtract: C = A − B.
4. Now repeat the same steps using C.

The amazing thing: no matter which number you start with, you always end up at 6174. And once you reach 6174, it stays there forever (because 7641 − 1467 = 6174 again).

Figure 3.5 below follows the steps starting from 6382.

Tap to zoom

Let’s do one full round together so the steps feel easy.

You do not need to know why 6174 is the magic number — even mathematicians find that surprising. But you can enjoy checking that it really works, with any starting number you like.

Even and odd patterns

Let’s remember an old friend.

Even and odd numbers follow tidy patterns when you add them. Watch:

• even + even = even (example: 4 + 6 = 10, even)
• odd + odd = even (example: 3 + 5 = 8, even)
• even + odd = odd (example: 4 + 5 = 9, odd)

Why does odd + odd give an even number? Picture each odd number as some pairs with one extra left over. When you add two odd numbers, the two leftover ones join up to make a new pair. Now everything is in pairs again, so the total is even. That is why, not just that.

Common Mistakes

Even fun topics have little traps. Here are the common ones.

Quick Check

Let’s see if the games make sense. Pick the best answer.

Practice Problems

Try each one yourself first. Then tap to check.

Easy

Medium

Challenge

Summary

• Numbers can describe an arrangement, not just count. In the line game, each child says how many of their neighbours are taller.
• The tallest person always says 0; an end person (with one neighbour) can only say 0 or 1.
• A supercell is a cell whose number is bigger than its neighbours. The largest number in a row is always a supercell; the smallest never is.
• To make the biggest number, put digits from large to small. For the smallest, put them small to large — but a number can never start with 0.
• The digit sum is the sum of all the digits. Different-sized numbers can share the same digit sum.
• A palindrome reads the same forwards and backwards (like 575). In reverse-and-add, you reverse a number and add until you get a palindrome.
• Kaprekar’s trick: from most 4-digit numbers, biggest − smallest, repeated, always reaches the magic number 6174.
• even + even = even, odd + odd = even, even + odd = odd — because two leftover ones from the odds always pair up.

What’s Next

You have seen how numbers hide patterns and play games. Next, you will learn how to collect numbers and show them clearly — using tally marks, tables, and simple pictures and bar graphs. That is the job of Chapter 4, Data Handling and Presentation. Once you can read and draw these, a big pile of numbers turns into a picture you can understand in one glance.