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I'm thinking of a number

Algebra · level 1, lesson 1 of 5

A question to start

"I'm thinking of a number from 0 to 10. Here are my clues.

  • It is more than 3.
  • It is less than 6.
  • It is not 4."

What is my number?

You do not have to read my mind. You only have to listen to the clues, one at a time, and think about which numbers are still possible. By the end of this lesson you will be a clue detective, and you will also know a secret: some clues help a lot, and some clues hardly help at all. (The answer is in the worked examples.)

The idea

A mystery number

A mystery number is a number that someone knows and you do not know yet. It is hidden, like a toy inside a closed box. You cannot see it, but it is a real number. It is not "any number": it is one number, and your job is to find out which one.

You find it from clues. A clue is a true thing about the mystery number. "It is more than 3" is a clue. "It is an even number" is a clue. "It is one more than 4" is a clue.

Grown-up mathematicians do this every day. They call the mystery number an unknown, and the whole subject of finding unknowns from clues is called algebra. So you are doing algebra already.

Each clue throws numbers out

Here is the most important trick. Before any clue, every number from 0 to 10 might be the mystery number. Put them all on a number line.

Now listen to a clue. Some numbers agree with the clue, and some do not. A number that does not agree cannot be the mystery number, so cross it out. Each clue crosses out some numbers. When only one number is left, you have found it.

Let's do the starting puzzle. The first clue is "more than 3". Numbers 0, 1, 2 and 3 are not more than 3, so they are out. The second clue is "less than 6". Numbers 6, 7, 8, 9 and 10 are not less than 6, so they are out.

012345678910 after "more than 3" and "less than 6", only 4 and 5 are left

Only 4 and 5 are left. Two numbers is not one number, so we need one more clue. The third clue says "not 4". Cross out 4, and only 5 is left. The mystery number is 5.

Why does crossing out work?

Because the mystery number is real, it must agree with every clue. A number that breaks even one clue cannot be it. So when we cross out the numbers that break a clue, we never cross out the mystery number by accident. It is always still there, hiding among the numbers that are left. That is why, when only one number is left, it must be the answer.

"One more than" and "one less than"

Some clues tell you the number almost straight away.

One more than 4 means: start at 4 and go one step up. On the number line, that is one hop to the right. You land on 5.

One less than 4 means: start at 4 and go one step down. That is one hop to the left. You land on 3.

012345678910 one more: hop right one less: hop left

"Two more than 4" means two hops to the right: 5, then 6. "Two less than 4" means two hops to the left: 3, then 2.

A good way to remember: the number line is like a staircase lying down. More is up the stairs, to the right. Less is down the stairs, to the left.

Checking a guess

Sometimes you just have a feeling: "I think it's 7!" A feeling is fine. A guess is a good way to start. But a detective never stops at a guess. A detective checks it.

To check a guess, go back to every clue and ask: "Does my guess agree with this clue?"

  • If the guess agrees with all of the clues, it is the answer.
  • If the guess breaks even one clue, it is wrong. But it was not a waste! It tells you something, and you can make a better guess.

Here is an example. Clues: "My number is more than 5. It is an even number. It is less than 8." You guess 7.

  • More than 5? Yes, 7 is more than 5. ✓
  • Even? No. Even numbers are the ones you can share into two equal piles: 2, 4, 6, 8, 10. 7 is not even. ✗
  • Less than 8? Yes. ✓

The guess 7 breaks a clue, so 7 is wrong. Which numbers are more than 5 and less than 8? Only 6 and 7. We just showed 7 is out, so try 6. More than 5 ✓, even ✓, less than 8 ✓. The number is 6.

The "too big, too small" game

Here is a game for two players. One player thinks of a number from 1 to 20. The other player guesses. After every guess, the first player says only "too big", "too small" or "yes!".

Each answer is a clue. If you guess 12 and hear "too big", then 12, 13, 14, all the way to 20 are out. If you guess 3 and hear "too small", then 1, 2 and 3 are out.

A clever way to play: guess a number in the middle of the numbers that are left. Then, whatever the answer is, about half of the numbers get crossed out. If you guess at the very end, like 1, and hear "too small", you only cross out one number. A middle guess is a strong clue-maker. Playing this way, you can always find a number from 1 to 20 in at most 5 guesses. Try it and see.

Not every clue is strong

Some clues cross out lots of numbers. Some cross out very few. "It is not 9" crosses out just one number. "It is less than 3" crosses out most of the numbers from 0 to 10. A clue like "it is a number" crosses out nothing at all, so it does not help.

And sometimes there are not enough clues. If the only clue is "it is more than 7" (from 0 to 10), then 8, 9 and 10 are all still possible. That is not a mistake. It just means the puzzle needs another clue before it has one answer.

Worked examples

1. The starting puzzle. From 0 to 10, the number is more than 3, less than 6, and not 4.

  • "More than 3" crosses out 0, 1, 2, 3.
  • "Less than 6" crosses out 6, 7, 8, 9, 10.
  • Left: 4 and 5. "Not 4" crosses out 4.
  • Left: 5. Check: 5 is more than 3 ✓, less than 6 ✓, not 4 ✓.

2. One more, one less. "My number is one more than 6." Start at 6 and hop one to the right: 7. Check: is 7 one more than 6? Count: six, seven. Yes ✓.

"My number is two less than 9." Start at 9 and hop two to the left: 8, then 7. Check: 7 and two more is 8, 9. Yes ✓. Notice that two different clues gave the same number. Many different clues can point to one number.

3. Using fingers. "My number is the number of fingers on one hand, and one more." One hand has 5 fingers. One more than 5 is 6. Hold up one whole hand and one finger on the other hand: 6 fingers.

4. A guess that is too big. "From 1 to 10, my number is less than 4, and it is not 1, and it is not 3." Sam guesses 5.

  • Less than 4? No, 5 is more than 4. ✗ So 5 is wrong.
  • Which numbers are less than 4? Only 1, 2, 3. Not 1, not 3: that leaves 2.
  • Check: 2 is less than 4 ✓, not 1 ✓, not 3 ✓.

5. The "too big, too small" game. The secret number is 14, from 1 to 20. Ana guesses in the middle each time.

Guess Answer Numbers still possible
10 too small 11 to 20
15 too big 11 to 14
12 too small 13, 14
13 too small 14
14 yes!

Five guesses, and each "too big" or "too small" threw out a lot of numbers.

Try it

  1. What number is one more than 7?
  2. What number is one less than 10?
  3. What number is two more than 5?
  4. From 0 to 10: my number is more than 6 and less than 8. What is it?
  5. From 0 to 10: my number is less than 5, it is even, and it is not 2. There are two numbers that fit. Which two? (Hint: is 0 even? Zero things can be shared into two equal piles of nothing.)
  6. Ben's clues are "more than 2" and "less than 5". He guesses 5. Check his guess. Is he right? If not, what numbers could it be?
  7. You are playing "too big, too small" from 1 to 10. You guess 5 and hear "too big". Then you guess 2 and hear "too small". Which numbers are still possible?
  8. Make up your own mystery number puzzle with three clues, so that only one number from 0 to 10 fits. Try it on someone at home.
Answers
  1. 8. Start at 7, one hop to the right: 8.
  2. 9. Start at 10, one hop to the left: 9.
  3. 7. Start at 5, two hops to the right: 6, then 7.
  4. 7. "More than 6" leaves 7, 8, 9, 10. "Less than 8" leaves only 7. Check: 7 is more than 6 ✓ and less than 8 ✓.
  5. 0 and 4. Less than 5 leaves 0, 1, 2, 3, 4. Even leaves 0, 2, 4. Not 2 leaves 0 and 4. Both fit every clue, so this puzzle needs one more clue, such as "it is not 0", to have just one answer.
  6. No. 5 is not less than 5 (it is the same as 5), so 5 breaks the second clue. The numbers more than 2 and less than 5 are 3 and 4. Ben needs another clue to choose between them.
  7. 3 and 4. "Too big" for 5 throws out 5, 6, 7, 8, 9, 10. "Too small" for 2 throws out 1 and 2. That leaves 3 and 4. A good next guess is either one of them.
  8. Many answers. One example: "More than 5. Less than 9. It is odd (it cannot be shared into two equal piles)." More than 5 and less than 9 leaves 6, 7, 8; odd leaves only 7. Check your own puzzle by crossing out numbers on a number line: exactly one should be left.

Common mistake

Stopping after one clue. If the clues are "more than 3" and "less than 6", some people hear "more than 3" and shout "4!". But 4 is only one of the numbers that fit the first clue. Listen to all the clues before you answer, and then check your answer against each one.

Mixing up "one more than 4" with 4. "One more than 4" is not 4. It is the number after 4, which is 5. Say it out loud and hop on the number line: "four... five".

Where it leads