Why a number trick always works

The idea

Letting a letter-number stand for any starting number and simplifying the expression after each step shows whether the result is the same for every number; algebra is thus used to justify general mathematical statements.

A trick with a prediction

Think of a number. Double it. Add four. Divide by two. Subtract the number you thought of. I predict you get 2.

Will everybody get 2, whatever number they thought of?

• Yes, always 2
• Only for some numbers
• No, it depends on the number

Run the trick on a bag

The bag holds your starting number. You never open it: its number is the letter x. Press the button for each step, then change the number in the bag and look at the last row.

The letter does the work

Letting a letter-number stand for any starting number and simplifying the expression after each step shows whether the result is the same for every number; algebra is thus used to justify general mathematical statements.

Let x be the number. After each step: Think of a number: x Double it: 2x Add four: 2x + 4 Divide by two: x + 2 Subtract the original number: x + 2 − x = 2

The x is gone, so the answer is 2 for every number. In Trick 2 the last step took away only 1, so x + 2 − 1 = x + 1: the letter stays and the answer depends on the number.

Two letters work too. In the date trick M is the month and D the day, and the steps end with 100M + 165 + D. Take away 165: the last two digits are D, the rest is M. For 291: 291 − 165 = 126, so M = 1 and D = 26.

Notes

Letting a letter-number stand for any starting number and simplifying the expression after each step shows whether the result is the same for every number; algebra is thus used to justify general mathematical statements.

Check yourself

Trick 2: think of a number, double it, add four, divide by two, subtract 1. What do you get?

Think of a number. Multiply by 3. Add 12. Divide by 3. Subtract the number you thought of. What is the answer, for every number?

Answer: 4

3x, then 3x + 12, then x + 4, then x + 4 − x = 4. Test it with 5: 15, 27, 9, 4.

Ananya follows the date trick and says 1167. On which day of the month was her date?

Answer: 2

1167 − 165 = 1002. The last two digits are 02, so D = 2, and what comes before is M = 10. It was 2 October.

Priya tries Trick 1 with 20 different numbers and always gets 2. What can show her it works for every number?

• Always 2, like Trick 1. Trick 1 subtracted the original number, and that removed the x. Subtracting 1 does not.
• Always 1. The x does not disappear. After dividing by two we have x + 2, and taking away 1 leaves x + 1.
• x + 1, so it depends on the number — correct. Yes! x + 2 − 1 = x + 1. The letter is still there, so different starting numbers give different answers.
• Trying 20 more numbers. However many numbers we try, we cannot try them all.
• A letter x for the starting number, simplifying after each step — correct. Yes! The letter stands for any number, so one calculation covers every number at once.
• Starting with a very big number. A big number is still only one number. It cannot show what happens to all the others.