Think-of-a-number tricks

Try this on a friend: "Think of any number. Double it. Add four. Divide by two. Subtract the number you started with." Then announce , with great confidence , "Your answer is 2." Your friend looks puzzled. Magic? No , algebra.

Concept

The trick works because every step you ask your friend to do is really an operation on an unknown. The result is forced by algebra to be the same regardless of the starting number.

Walking through it. Let your friend's secret number be $x$. Apply each step symbolically:

1. Think of $x$.
2. Double it: $2x$.
3. Add four: $2x + 4$.
4. Divide by two: $\dfrac{2x + 4}{2} = x + 2$.
5. Subtract the original number: $(x + 2) - x = 2$.

The $x$ cancels , the final value is always $2$, no matter what number was chosen. That cancellation is the heart of every such trick.

Reverse-engineering a trick. If you instead want the final answer to be 3, then in step $3$ ask them to "Add six" , because $\dfrac{2x + 6}{2} - x = 3$. For final answer $5$, ask "Add ten". In general, to make the trick end at value $k$, ask them to "Add $2k$" at step $3$.

A bigger trick , the calendar guess. Ask a friend to do these steps with a chosen date (a month $M$ and day $D$).

1. Multiply the month by $5$: $5M$.
2. Add $6$: $5M + 6$.
3. Multiply by $4$: $20M + 24$.
4. Add $9$: $20M + 33$.
5. Multiply by $5$: $100M + 165$.
6. Add the day: $100M + 165 + D$.

If the friend says "$291$," subtract $165$: $291 - 165 = 126 = 100M + D$. Since $D$ is at most $31$ (two digits), the last two digits of $126$ are $D = 26$ and the rest is $M = 1$. So the date is January $26$th.

Inventing your own. Any sequence of operations whose net effect on $x$ ends in a constant works. Some safe building blocks:

• "Double, add a number, halve, subtract the original" , collapses to half the number you added.
• "Multiply by $3$, add a number, subtract twice the original" , collapses to (original + the constant).

Why this is powerful. Once you can write a procedure with $x$, you have a way to explain, predict, and modify it. That is the everyday power of algebra.

Worked examples

Example 1. Trick: "Triple it, add six, divide by three, subtract the original." Verify the final answer is always $2$.

• Let $x$ be the number. $3x \to 3x+6 \to \dfrac{3x+6}{3} = x+2 \to (x+2) - x = 2$. ✓

Example 2. What number should replace "add four" in the original trick so the final answer is always $7$?

• $\dfrac{2x + k}{2} - x = \dfrac{k}{2}$. For final $= 7$, $k = 14$. So ask to "add $14$".

Example 3. A friend chose a date, did the steps in the calendar trick, and got $1390$. Find the date.

• $1390 - 165 = 1225 = 100M + D$. Last two digits: $D = 25$. Other digits: $M = 12$. The date is December $25$.

Example 4. Trick: "Pick a number. Add $3$. Multiply by $2$. Subtract $4$. Divide by $2$. Subtract the original." Predict the constant.

• $x \to x+3 \to 2x+6 \to 2x+2 \to x+1 \to (x+1) - x = 1$. The answer is always $1$.

Try it yourself

1. Verify the trick "Think of $x$. Multiply by $4$. Add $8$. Divide by $4$. Subtract the original." Final answer?
2. Adjust the trick "Double, add $k$, halve, subtract the original" so the final answer is $10$. Find $k$.
3. Calendar trick: a friend reports $269$. Find the date.
4. Make up a trick where the final answer is always $0$.
5. Trick: "$x \to 5x \to 5x + 20 \to (5x+20)/5 \to \text{subtract } x$." Final answer?
6. Calendar trick: a friend reports $394$. Find the date.
7. Trick: "Pick $x$. Multiply by $6$. Add $12$. Divide by $3$. Subtract $2x$." Final?
8. Why does cancelling the $x$ guarantee the trick "always" works?

Activity

Trick designer. Invent your own trick whose final answer is your favourite single-digit number. Write it out using $x$ to be sure the $x$'s cancel correctly. Test it on three friends. If any of them gets a different answer, find the slip and fix it.