Skill: Use logical elimination

When several answers seem possible, use logical elimination:

1. List all the possible answers.
2. Use the first condition to cross out answers that do not fit.
3. Use the next condition on the answers left.
4. Keep going until every remaining answer fits all the conditions.
5. Check the final answer. An answer that is “not ruled out yet” is not proved until it passes every condition.

Example

A secret whole number is between $1$ and $(20)$.

• It is even.
• It is greater than $(10)$.
• It is less than $(16)$.
• It is not $(12)$.

What is the secret number?

Step 1: List the possibilities

The number could be any of:

$$1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20$$

Step 2: Use “even”

Cross out the odd numbers. The numbers left are:

$$2,4,6,8,10,12,14,16,18,20$$

Step 3: Use “greater than $(10)$”

Cross out $(10)$ and the smaller numbers:

$$12,14,16,18,20$$

Step 4: Use “less than $(16)$”

Cross out $(16)$ and the larger numbers:

$$12,14$$

Both numbers still fit the conditions used so far. They are possible, but neither is proved yet.

Step 5: Use “not $(12)$”

Cross out $(12)$. The only number left is:

$$\boxed{14}$$

Check: $(14)$ is even, greater than $(10)$, less than $(16)$, and not $(12)$. Therefore, the secret number is $(14)$.