Algebra Explains Why the Number Trick Never Fails

Because the starting number cancels out along the way. Write the steps with a letter instead of a number and you can watch it disappear, leaving only a fixed amount behind.

That is what algebra is for here — not finding an unknown, but proving something about every possible number at once. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: modelling a number trick, completing a number pyramid, and expressing its top in terms of its base.

Call the chosen number $x$, write each instruction as an operation on $x$, then simplify.

Trick 1. Think of a number. Double it. Add $10$. Halve the result. Subtract your original number.

Following it with $x$:

- Think of a number: $x$
- Double it: $2x$
- Add 10: $2x+10$
- Halve it: $\frac{2x+10}{2}=x+5$
- Subtract the original: $(x+5)-x=5$

The answer is always 5, whatever was chosen. Testing with $x=7$: $7\to 14\to 24\to 12\to 5$. Testing with $x=100$: $100\to 200\to 210\to 105\to 5$.

Trick 2. Think of a number. Add $3$. Multiply by $2$. Subtract $4$. Halve it. Subtract your original number.

$$x\to x+3\to 2x+6\to 2x+2\to x+1\to 1$$

The answer is always 1. Testing with $x=7$: $7\to 10\to 20\to 16\to 8\to 1$.

Trick 3. Think of a number. Multiply by $3$. Add $12$. Divide by $3$. Subtract the original.

$$x\to 3x\to 3x+12\to x+4\to 4$$

Always 4.

The pattern is visible once written this way: the **$x$ term is removed by the final subtraction**, and what survives is whatever the constants worked out to. In Trick 1 the $10$ was halved to $5$; in Trick 3 the $12$ was divided by 3 to give $4$ — so you can predict a trick's answer by following only the constants.

In a number pyramid each cell is the sum of the two cells directly below it. Work upward, and use letters where the base entries are unknown.

With numbers. Base $2,5,3$:

- Middle row: $2+5=7$ and $5+3=8$
- Top: $7+8=15$

With numbers again. Base $4,1,6$:

- Middle row: $5$ and $7$
- Top: $12$

With variables. Base $a,b,c$:

- Middle row: $a+b$ and $b+c$
- Top: $(a+b)+(b+c)=a+2b+c$

Working downward instead. If the top is $20$ and the middle row is $8$ and $12$, then the base entries must satisfy $a+b=8$ and $b+c=12$. With $b=5$, we get $a=3$ and $c=7$ — and checking, the base $3,5,7$ gives a middle row of $8,12$ and a top of $20$.

A four-wide base. Base $a,b,c,d$:

- Row 3: $a+b$, $b+c$, $c+d$
- Row 2: $(a+2b+c)$ and $(b+2c+d)$
- Top: $a+3b+3c+d$

Testing with base $1,2,3,4$: row 3 is $3,5,7$; row 2 is $8,12$; top is $20$. And the formula gives $1+6+9+4=20$. They agree.

The reason the middle entries count more is that they feed into two cells above them, while the outer entries feed into only one. That is why $b$ appears twice in $a+2b+c$ — and it explains why changing a middle base number moves the top far more than changing an end one.

Substitute the numerical base values into the algebraic expression and check it matches the pyramid you built by adding.

Three-wide base, where the top is $a+2b+c$:

Base $2,5,3$. Building upward gave a top of $15$. The formula gives

$$2+2(5)+3=2+10+3=15$$

They agree.

Base $4,1,6$, which gave a top of $12$:

$$4+2(1)+6=12$$

Agrees.

Base $7,0,7$:

$$7+0+7=14$$

and building upward gives middle row $7,7$ and top $14$. Agrees.

Four-wide base, where the top is $a+3b+3c+d$:

Base $1,2,3,4$, which gave a top of $20$:

$$1+3(2)+3(3)+4=1+6+9+4=20$$

Agrees.

Base $2,1,1,2$:

$$2+3+3+2=10$$

and building upward gives row 3 as $3,2,3$, row 2 as $5,5$, and top $10$. Agrees.

The coefficients follow a pattern worth noticing: $1,2,1$ for a three-wide base and $1,3,3,1$ for a four-wide one. Those are the same numbers that appear when expanding $(a+b{)}^2$ and $(a+b{)}^3$, and they arise for the same reason — counting how many routes lead from each base cell up to the top.

Verifying with two different bases is what makes the check meaningful. A formula that matches one example might do so by accident, but one that matches several, including a base containing a zero, is almost certainly right — which is exactly the habit the syllabus is building.