13.2 Thinking About ‘Think of a Number’ Tricks

In Grade 7 you ran into “think of a number” tricks. Try this one with whatever number you fancy.

  1. Think of a number.
  2. Triple it.
  3. Add twelve.
  4. Divide by three.
  5. Take away the number you first thought of.

My guess is you finish on 4. Did I get it right? Test it with 7, with 40, with \tfrac12, the answer is 4 every single time. How can a trick name your result before you’ve even worked it out?

The trick to the trick is to walk through the steps not with one fixed number but with a letter-number x that stands for whatever you picked.

Worked Example

Why the trick always lands on 4.

Step In words In algebra
1 Think of a number x
2 Triple it 3x
3 Add twelve 3x + 12
4 Divide by three \dfrac{3x+12}{3} = x + 4
5 Take away the original (x+4) - x = 4

The x wipes itself out at the final step, so the answer is 4 no matter what x was. That is the entire secret.

The Key Idea A number trick “works for everybody” precisely when the chosen number cancels out of the closing expression. Algebra lets us watch that cancellation unfold, and lets us arrange it deliberately.

Math Talk How would you tweak the steps so the answer always comes out 6? What about 10? (Hint: here “+12” turned into “÷3 → +4”. Which number should you add at step 3 to be left with 6?) Can you string together a longer set of steps that still funnels everyone to one fixed value?

Use the checker below to type in your own steps and watch whether the chosen number cancels.

Think-of-a-Number Tester

Pick any two starting numbers. If the trick truly works, both runs land on the same value.

  
Steps: triple, add twelve, divide by three, subtract the original number.

Worked Example

Design a trick that always finishes on 6.

Work backwards. We want the closing step “subtract the original x” to be applied to something of the form x + 6, since (x+6) - x = 6. To reach x + 6 after a “divide by three”, the line before division must read 3x + 18 (because \tfrac{3x+18}{3} = x + 6). So:

Step In words In algebra
1 Think of a number x
2 Triple it 3x
3 Add eighteen 3x + 18
4 Divide by three x + 6
5 Subtract the original 6

Replacing “add twelve” by “add eighteen” shifts the answer from 4 to 6. Inventing a trick is just running the algebra in reverse.

A trick that finds a date

Here is a livelier one. Aarav asks Diya to picture a date and keep it to herself. She settles on Independence Day, 15 August, written as day 15, month 8.

Step Instruction Diya’s working
1 Think of a date 15/08
2 Multiply the month by 5 8\times 5 = 40
3 Add 6 40 + 6 = 46
4 Multiply by 4 46 \times 4 = 184
5 Add 9 184 + 9 = 193
6 Multiply by 5 193 \times 5 = 965
7 Add the day 965 + 15 = 980

Diya announces 980, and Aarav shoots straight back, “You were thinking of 15 August!” How does he do it?

Let the month be M and the day be D, and run the same steps in algebra:

\begin{aligned} 5M \;\xrightarrow{+6}\; 5M+6 \;&\xrightarrow{\times 4}\; 20M+24 \;\xrightarrow{+9}\; 20M+33\\ &\xrightarrow{\times 5}\; 100M+165 \;\xrightarrow{+D}\; 100M + 165 + D. \end{aligned}

So the final answer is always 100M + 165 + D. Aarav simply subtracts 165:

980 - 165 = 815 = 100M + D.

Since the day D is never more than 31, only two digits, the last two digits of 815 are the day and whatever sits before them is the month. Here M = 8 and D = 15: the 15th of August.

Worked Example

Diya’s next answer is 1395. Which date?

Subtract 165: \;1395 - 165 = 1230 = 100M + D. The last two digits give D = 30, and what remains is M = 12. The date is the 30th of December.

Worked Example

A friend reports 379. Find the date, and the catch.

Subtract 165: \;379 - 165 = 214 = 100M + D. The last two digits give D = 14, and the rest is M = 2. The date is 14 February. Notice the answer 379 is only three digits because the month is single-digit; the recipe still works because the “100M” part keeps the day safely parked in the last two places.

Try This Recover the date behind each final answer.  (i) 1170  (ii) 487  (iii) 996

Then test the trick on a friend, asking them to start from their birthday.

Subtract 165 from each answer; the last two digits are the day D, and the rest is the month M.

    1. 1170 - 165 = 1005 \Rightarrow M = 10,\ D = 05: 5 October.
    1. \;487 - 165 = 322 \Rightarrow M = 3,\ D = 22: 22 March.
    1. \;996 - 165 = 831 \Rightarrow M = 8,\ D = 31: 31 August.

Math Talk Can you reshuffle the steps and still get the date back? If you change step 3 or step 5, the number you subtract at the end may no longer be 165. Work out what it should become, that calculation is the act of inventing your own trick.