Context: Consider the following 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. (This trick always results in ). Q. Can you come up with more complicated steps that always lead to the same final value?

Understand the Question Number tricks work by performing a series of algebraic operations on a variable, say , such that the variable cancels out in the end, leaving a constant value. In the given trick, starting with yields . To create a more complicated trick, we can introduce additional intermediate operations (like extra additions, multiplications, and subtractions of multiples of ) designed so that the final simplification still eliminates and results in . Step 1 · Analyze the Original Trick Let the chosen number be . Following the given steps: 1. Double the number: 2. Add four: 3. Divide by two: 4. Subtract the original number : Step 2 · Design and Verify New Steps We can design a set of steps that also simplifies to : 1. Think of a number. 2. Multiply it by six. 3. Add twelve to the result. 4. Subtract two times your original number. 5. Divide the new result by four. 6. Subtract one from this value. 7. Subtract the original number you thought of. Verification using algebra: Let the number be . Multiply by six: Add twelve: Subtract two times the original number: Divide by four: Subtract one: Subtract the original number : Answer > Yes. A set of steps that always leads to is: 1. Think of a number. 2. Multiply it by six. 3. Add twelve. 4. Subtract two times your original number. 5. Divide the result by four. 6. Subtract one. 7. Subtract your original number. Common Mistakes Failure to Cancel the Variable: Forgetting to subtract the original number (or appropriate multiples of it), leaving the final answer dependent on the chosen number . Division Distributive Error: Dividing only the constant or only the variable term (e.g., writing as instead of ).

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Question 3

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?

Solution

Understand the Question

Number tricks work by performing a series of algebraic operations on a variable, say $x$ , such that the variable cancels out in the end, leaving a constant value.
In the given trick, starting with $x$ yields $(2x + 4)/2 - x = (x + 2) - x = 2$ .
To create a more complicated trick, we can introduce additional intermediate operations (like extra additions, multiplications, and subtractions of multiples of $x$ ) designed so that the final simplification still eliminates $x$ and results in $2$ .

Step 1 · Analyze the Original Trick

Let the chosen number be $x$ .

Following the given steps:

Double the number: $2x$
Add four: $2x + 4$
Divide by two:

\begin{aligned} \dfrac{2x + 4}{2} &= \dfrac{2x}{2} + \dfrac{4}{2} \\[0.6em] &= x + 2 \end{aligned}

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

Step 2 · Design and Verify New Steps

We can design a set of steps that also simplifies to $2$ :

Think of a number.
Multiply it by six.
Add twelve to the result.
Subtract two times your original number.
Divide the new result by four.
Subtract one from this value.
Subtract the original number you thought of.

Verification using algebra:

Let the number be $x$ .

Multiply by six: $6x$
Add twelve: $6x + 12$
Subtract two times the original number: $(6x + 12) - 2x = 4x + 12$
Divide by four:

\begin{aligned} \dfrac{4x + 12}{4} &= \dfrac{4x}{4} + \dfrac{12}{4} \\[0.6em] &= x + 3 \end{aligned}

Subtract one: $(x + 3) - 1 = x + 2$
Subtract the original number $x$ : $(x + 2) - x = 2$

Answer

Yes. A set of steps that always leads to $2$ is:

Think of a number.
Multiply it by six.
Add twelve.
Subtract two times your original number.
Divide the result by four.
Subtract one.
Subtract your original number.

Common Mistakes

Failure to Cancel the Variable: Forgetting to subtract the original number (or appropriate multiples of it), leaving the final answer dependent on the chosen number $x$ .
Division Distributive Error: Dividing only the constant or only the variable term (e.g., writing $\dfrac{4x + 12}{4}$ as $4x + 3$ instead of $x + 3$ ).

Question 3

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