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Play: Mystery numbers and balance scales

Algebra · step 1 of 5

A question to start

"I'm thinking of a number. I add 4 to it, and I get 11. What is my number?"

You probably said 7 quite quickly. But how did you know? You did not try every number. You thought something like: the number was 4 less than 11, and 11 take away 4 is 7. You worked backwards. That one move, undoing what was done to a mystery number, is the heart of a whole branch of mathematics called algebra.

The idea

Mystery numbers

A mystery number puzzle tells you what happened to a hidden number and what came out at the end. Your job is to find what went in. There are two good ways to do it.

Way 1: work backwards

Every action has an opposite that undoes it.

If the puzzle says ... ... undo it by
add 4 taking away 4
take away 3 adding 3
double it (times 2) halving it (divide by 2)
divide by 5 multiplying by 5

To find the mystery number, start at the end and undo each action, going in the opposite order. It is like taking off your shoes and socks: you put your socks on first, but you take your shoes off first.

Way 2: keep the scale balanced

Picture a see-saw scale. On the left pan is a closed bag with a mystery number of marbles inside, and 3 loose marbles. On the right pan are 7 loose marbles. The scale is balanced: both sides weigh the same.

?

How many marbles are in the bag? Here is the golden rule of the balance:

Whatever you do to one side, do the same to the other side, and the scale stays balanced.

Take 3 marbles off the left. To keep it balanced, take 3 marbles off the right too. Now the bag sits alone on the left, and 4 marbles sit on the right. The scale still balances, so the bag holds 4 marbles.

Always check

When you think you have found the mystery number, put it back into the puzzle and see if it works. The bag holds 4: 4 marbles and 3 more make 7. Yes! Checking takes a few seconds and catches most mistakes.

Worked example

The puzzle: "I think of a number. I double it, then add 1, and I get 15. What is my number?"

  1. Write the steps in order. Start with the mystery number, then double it, then add 1, and the result is 15.
  2. Undo them in the opposite order. The last thing done was add 1, so undo it first: 15 take away 1 is 14.
  3. Undo the next one. Before that the number was doubled, so halve: half of 14 is 7.
  4. Answer. The mystery number is 7.
  5. Check. Double 7 is 14, and add 1 is 15. It works.

The same puzzle on a balance. On the left: two identical bags (the number doubled) and 1 marble. On the right: 15 marbles. Take 1 marble off each side: two bags balance 14 marbles. Two equal bags balance 14, so share the 14 into two equal piles: each bag balances 7. Same answer, a different way to see it.

A mind-reading trick

Try this on a friend. "Think of a number. Double it. Add 10. Halve the answer. Now take away the number you first thought of." Then say, very slowly, as if reading their mind: "Your answer is ... 5."

It always works, whatever number they choose. Why? Imagine their number as a bag. Doubling gives two bags. Adding 10 gives two bags and 10 marbles. Halving gives one bag and 5 marbles. Taking away the number they thought of takes away the bag. Five marbles are left, every time. In the next step you will write this trick in a short, neat way.

Try it

  1. I think of a number and take away 6. I get 9. What is my number?
  2. I think of a number and multiply it by 4. I get 28. What is my number?
  3. I think of a number, double it, then take away 3, and I get 11. What is my number?
  4. A balanced scale has three equal bags on one side and 12 marbles on the other. How many marbles are in each bag?
  5. A balanced scale has one bag and 5 marbles on the left, and 2 marbles and 9 marbles on the right. How many marbles are in the bag?
  6. Change the mind-reading trick so that the answer is always 8. (Hint: what should you add?)
Answers
    1. Undo "take away 6" by adding 6: 9 and 6 make 15. Check: 15 take away 6 is 9.
    1. Undo "times 4" by dividing by 4: 28 shared into 4 is 7. Check: 7 times 4 is 28.
    1. Undo "take away 3" first: 11 and 3 make 14. Then undo "double": half of 14 is 7. Check: double 7 is 14, take away 3 is 11.
  4. 4 in each bag. Three equal bags balance 12, so share 12 into three equal piles: 4 each.
    1. The right side holds 2 and 9, which is 11 marbles. Take 5 off both sides: the bag balances 6. Check: 6 and 5 make 11.
  6. Add 16 instead of 10. Two bags and 16 marbles, halved, is one bag and 8 marbles; take away the bag and 8 are left. (Whatever you add, the answer is half of it.)

Common mistake

Undoing in the wrong order. In "double it, then add 1, and get 15", some people halve first: half of 15 is 7 and a half, take away 1 is 6 and a half. Check it: double 6 and a half is 13, add 1 is 14, not 15. Wrong! The last thing done must be the first thing undone, just like shoes before socks.

Where it leads