Context: Think of a number trick: 1. Think of a number. 2. Double it. 3. Add four. 4. Divide by two. 5. Subtract the original number you thought of. Q. I predict you get 2. Am I right? Try it out with different starting numbers. Do you always end up with the same value, 2? Why?

Algebra Play | IT

Question 1

Context: Think of a number trick:

Think of a number.
Double it.
Add four.
Divide by two.
Subtract the original number you thought of.

Q. I predict you get 2. Am I right? Try it out with different starting numbers. Do you always end up with the same value, 2? Why?

Solution

We can use algebra to represent the unknown number and see what happens after each step of the trick.

Step 1 — Trying with examples

Let us try the trick with a few different numbers. First, let us pick the number 5.

Think of a number: $5$
Double it: $5 \times 2$

$= 10$

Add four: $10 + 4$

$= 14$

Divide by two: $14 \div 2$

$= 7$

Subtract the original number: $7 - 5$

$\boxed{\textbf{2}}$

Now, let us try with the number 10.

Think of a number: $10$
Double it: $10 \times 2$

$= 20$

Add four: $20 + 4$

$= 24$

Divide by two: $24 \div 2$

$= 12$

Subtract the original number: $12 - 10$

$\boxed{\textbf{2}}$

It seems the prediction is right for these examples.

Step 2 — Using algebra

Let us use a variable to represent any number. Let $x$ be the number you think of.

Think of a number: $x$
Double it: $x \times 2$

$= 2x$

Add four: $2x + 4$
Divide by two: $\frac{2x + 4}{2}$

$= \frac{2x}{2} + \frac{4}{2}$

$= x + 2$

Subtract the original number: $(x + 2) - x$

$= x - x + 2$

$\boxed{\textbf{2}}$

The result is always 2, regardless of the starting number $x$ . This is because the operations cancel out the original number.

Answer

(i) Yes, your prediction is right. (ii) Yes, we always end up with the same value, 2. (iii) The algebraic steps show that the original number cancels out, leaving only 2.

More questions in IT

Context: Think of a number trick:

Think of a number.
Double it.
Add four.
Divide by two.
Subtract the original number you thought of.

Q. I predict you get 2. Am I right? Try it out with different starting numbers. Do you always end up with the same value, 2? Why?

Context: Consider the following number trick:

Think of a number.
Double it.
Add four.
Divide by two.
Subtract the original number you thought of. (This trick always results in 2).

Q. How would you change this game to make the final answer 3? What about 5?

Context: Consider the following number trick:

Think of a number.
Double it.
Add four.
Divide by two.
Subtract the original number you thought of. (This trick always results in 2).

Q. Can you come up with more complicated steps that always lead to the same final value?

Context: In the date trick, the final answer is given by $100M + 165 + D$ , where $M$ is the month and $D$ is the day.

Q. Find the dates if the final answers are the following:

(i) 1269 (ii) 394 (iii) 296

Context: In the date trick, the final answer is given by $100M + 165 + D$ , where $M$ is the month and $D$ is the day.

Q. Can you change the steps in this trick and still find the original date? Instead of subtracting 165 from the final answer, you might have to subtract some other number.

Try to devise your own 'Think of a Number' trick.

Use the same rule to fill these pyramids:

Fill the following pyramids:

What is the relationship between the numbers in the bottom row and the number at the top?

Let us start with the simplest pyramid.

Q10

What about a pyramid with three rows?

Using letter numbers for the bottom row, we can write an expression for the top row.

Q11

In the following grids, find the values of the shapes and fill in the empty squares:

Q12

Context: 6.5 The Largest Product

Q. Fill the digits 2, 3, and 5 in $\square\square \times \square$ , using each digit once. What is the largest product possible?

Q13

Context: Mukta's trick: Choose a 2-digit number of different digits, reverse the digits to get another number, find their difference, and divide the result by 9.

Q. If we choose other 2-digit numbers, and follow the steps, will there always be no remainder?

Q14

Context: Suppose a two-digit number is $ab$ . When it is reversed, the new number is $ba$ . If $b > a$ , the difference is $ba - ab = 9(b - a)$ , which is divisible by 9.

Q. Can you work out what happens if $a > b$ ?