Algebra Play

Why This Matters

Have you ever seen someone do a “magic” number trick? They say: “Think of any number. Don’t tell me. Double it. Add four. Halve it. Take away the number you started with.” Then they grin and announce: “Your answer is 2.”

And it always is. Every time. For any number you pick.

It feels like mind-reading. But it isn’t magic at all. It is algebra quietly doing its job behind the curtain.

In Class 6 and 7 you used algebra to stand in for unknown numbers. Now we get to play with it. We will pull these tricks apart and see the gears inside. We will learn why they always work, not just that they work. Once you see how the machine runs, you can build your own tricks to amaze your friends.

This is also the deeper point of algebra. A single letter, like x, can stand for every number at once. So when we prove something with x, we have proved it for the whole infinite list of numbers in one short line. That power shows up everywhere in maths, science, and beyond.

The Big Idea

A number trick uses a hidden number you never tell anyone. If we call that hidden number x and follow the steps, the x carries through the whole chain. At the end, the x either cancels out (so the answer is fixed) or it can be pulled back out (so we can find it). Letters turn a “magic” trick into something we can fully explain. The same letters let us form equations and solve them, fill number pyramids and grids, and prove that a trick must work for every number, not just the ones we tested.

Let’s Break It Down

The ‘Think of a Number’ trick, traced with a letter

Let us take the trick from the start, step by step. We will do it two ways at once: with a real number, say 7, and with the letter x standing for “whatever you chose”.

1. Think of a number. Real: 7. Letter: x.
2. Double it. Real: 14. Letter: 2x.
3. Add four. Real: 18. Letter: 2x + 4.
4. Divide by two. Real: 9. Letter: x + 2. (Half of 2x is x; half of 4 is 2.)
5. Subtract the starting number. Real: 9 − 7 = 2. Letter: (x + 2) − x = 2.

Look at the last line. The x and the −x cancel. They wipe each other out. What is left is just 2.

That is the whole secret. The answer does not depend on x at all, because the x disappears. So whatever you start with, you land on 2. The picture below traces both columns side by side so you can watch the x vanish.

Tap to zoom

Now you can see how to change the trick. The “add four” step became ”+ 2” after halving, and that 2 is the final answer. Want the answer to be 3 instead? Then you need ”+ 3” to survive at the end, so add six instead of four (because half of 6 is 3). Want the answer to be 5? Add ten. You are in control once you see the algebra.

Forming an expression — the recipe view

Each instruction in the trick adds one more layer to an expression. “Double it” multiplied by 2. “Add four” added 4. Step by step, the expression 2x + 4 is built up.

An expression is just a recipe written in symbols. It does not have an equals sign yet. It is waiting for you to put in a value for x. Figure 6.2 shows this building-up as a tree you read from the bottom to the top.

Tap to zoom

Reading it as a recipe makes algebra less scary. 2x + 4 simply means: “take your number, double it, then add four.” Nothing more.

Forming and solving an equation

So far we only built expressions. The fun jumps up when we get an equation — an expression set equal to something.

Here is a small puzzle. “I doubled a number and added four, and got ten. What was my number?” In symbols: 2x + 4 = 10.

Now we must solve it. Solving means finding the value of x that makes both sides truly equal.

The safest way to picture this is a balance scale. The equals sign means the two sides weigh the same — the scale is level. The golden rule is simple: whatever you do to one side, do exactly the same to the other side. Then the scale stays level and the equation stays true.

Let us see this on the scale before we write the steps.

Tap to zoom

Now the same thing written as steps.

Notice the order. To free x, we peeled off the steps in reverse. The expression added 4 last, so we removed 4 first. The expression doubled first, so we undid the doubling last. Solving an equation is like untying a knot — you undo the last loop first.

Number pyramids

Here is a different playground for the same idea. In a number pyramid, each block is the sum of the two blocks directly below it. So if the bottom two blocks are 4 and 9, the block above them is 4 + 9 = 13.

Filling a pyramid upward is easy — you just add. The puzzle gets interesting when you know the top but not the bottom. Then algebra rescues you: put a letter in an empty block, use the “sum of the two below” rule to write equations, and solve.

Figure 6.4 shows both the general rule and a real pyramid solved this way.

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Let us walk through that solve carefully.

There is also a lovely pattern hiding here. For the simplest pyramid, the bottom blocks a and b give a top of a + b. For a 3-row pyramid with bottom a, b, c, the top is a + 2b + c — the middle block is counted twice. Knowing the pattern lets you find the top without building the whole pyramid.

The calendar grid trick

Open any month on a calendar. Ask a friend to pick a 2 × 2 square of four dates and tell you only their sum. From that one number, you can name all four dates. How?

Look at how the four numbers relate. If the top-left date is a, then the date to its right is a + 1 (the next day). The date directly below is a + 7 (one week later). The bottom-right is a + 8. So the grid is a, a + 1, a + 7, a + 8 — every box is written in terms of one letter.

Add them up: a + (a + 1) + (a + 7) + (a + 8) = 4a + 16. So the friend’s sum is always 4a + 16.

The trick feels like magic to your friend, but you simply solved 4a + 16 = 36. Change the shape of the grid and you get a different expression — and a new trick of your own.

Decoding the date trick

Some tricks hide two numbers, not one, yet algebra still untangles them. Here is the classic “guess my date” trick. Tell a friend:

• Multiply the month by 5. Add 6. Multiply by 4. Add 9. Multiply by 5. Finally, add the day. Tell me the result.

From that one number you announce their exact date. Let us see why, with the month as M and the day as D.

Following the steps: 5M, then 5M + 6, then 20M + 24, then 20M + 33, then 100M + 165, then finally 100M + 165 + D.

Notice that every step except the last only touches M. So the leftover 165 is the same every single time. Strip it off and the rest falls apart neatly. Figure 6.5 shows the full reveal for the answer 291.

Tap to zoom

Why divisibility tricks work

Now a trick that always lands on a multiple of 9. Pick any 2-digit number with two different digits — say 47. Reverse the digits to get 74. Subtract the smaller from the bigger: 74 − 47 = 27. Divide by 9: you get 3, with no remainder. It works for any such number. Why?

The key idea is how a 2-digit number is really built. The number “ab” is not a times b. It means 10a + b — that is, the tens digit counts for ten each, plus the ones digit. So 47 is 10 × 4 + 7. Figure 6.6 uses this to crack the trick.

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Read the working: the difference is 9b − 9a = 9(b − a). That is 9 multiplied by a whole number. Anything that is 9 times a whole number is divisible by 9. So there can never be a remainder. The algebra proves it for every 2-digit number in one short line — something no amount of testing single examples could ever fully guarantee.

Inventing your own trick

Now you are ready to be the magician. To build a “think of a number” trick, chain together steps and make sure the x behaves the way you want.

• To make the answer always the same, design the steps so the x cancels at the end (like our +x and −x). The leftover constant is your answer.
• To make a trick where you can guess the number back, keep the x alive but tucked inside, then undo each step in reverse to recover it.

Common Mistakes

These are the slips students make most often with algebra play. Read them once and you will sidestep them.

Quick Check

Try these. Each one checks an idea from the chapter.

Practice Problems

Try each one yourself first. Only then tap to see the full solution.

Easy

Medium

Challenge

Summary

• A ‘think of a number’ trick works because the steps make the hidden number x cancel out (so the answer is fixed) or let us pull it back out. Algebra reveals the secret.
• An expression like 2x + 4 is a recipe in symbols, built one step at a time. It has no equals sign yet.
• An equation sets an expression equal to something. To solve it, picture a balance scale and do the same thing to both sides until the letter stands alone — peeling off the steps in reverse order.
• In a number pyramid, each block is the sum of the two below it. Putting a letter in an empty block turns the puzzle into equations you can solve.
• The calendar grid and date tricks write every box in terms of one or two letters, so one reported number is enough to find them all.
• A 2-digit number “ab” means 10a + b, not a × b. This place value is why reversing digits and subtracting always gives 9(a − b) — a multiple of 9.
• Above all: algebra proves a trick works for every number at once, where testing examples never can. That is its real power.

What’s Next

You have seen how a single letter can stand for every number at once, explain “magic” tricks, and prove statements for all numbers in one line. That habit of using letters to capture a whole pattern is the heart of algebra.

Next we turn that power onto shapes and space. The next chapter is Area. There you will measure the space inside flat figures — rectangles, triangles, and more — and you will see algebra return to give you neat formulas for them. See you there!