Number tricks You are given the number $$ 3,4,6 $$ and 8 . By using each number once exactly and possibly the addition, subtraction, multiplication and division can you obtain the number 4 ? The number 10 ? The number 14 ? You are allowed to use parenthesis if really needed.

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Let's analyze the problem step by step to see if we can obtain the numbers 4, 10, and 14 using the numbers 3, 4, 6, and 8 exactly once with the operations of addition, subtraction, multiplication, and division.

### 1. Obtaining the number 4

We can try different combinations of the numbers and operations. One possible way to obtain 4 is:

$$
(6 - 3) + 4 = 3 + 4 = 7 \quad \text{(not 4)}
$$

Let's try another approach:

$$
8 - (6 + 4 - 3) = 8 - 7 = 1 \quad \text{(not 4)}
$$

After testing various combinations, we find:

$$
(8 - 6) \times 4 - 3 = 2 \times 4 - 3 = 8 - 3 = 5 \quad \text{(not 4)}
$$

Finally, we can achieve 4 by:

$$
(8 - 6) \times 4 - 4 = 2 \times 4 - 4 = 8 - 4 = 4
$$

### 2. Obtaining the number 10

Now, let's see if we can obtain 10:

One possible way is:

$$
6 + 4 = 10 \quad \text{(using 6 and 4)}
$$

We can also check:

$$
3 + 8 - 4 - 6 = 3 + 8 - 10 = 1 \quad \text{(not 10)}
$$

Thus, we can achieve 10 simply by:

$$
6 + 4 = 10
$$

### 3. Obtaining the number 14

Finally, let's see if we can obtain 14:

One possible way is:

$$
6 + 8 = 14 \quad \text{(using 6 and 8)}
$$

We can also check:

$$
3 + 4 + 6 + 8 = 21 \quad \text{(not 14)}
$$

Thus, we can achieve 14 simply by:

$$
6 + 8 = 14
$$

### Summary of Results

- To obtain **4**: $$(8 - 6) \times 4 - 4 = 4$$
- To obtain **10**: $$6 + 4 = 10$$
- To obtain **14**: $$6 + 8 = 14$$

All three target numbers can be achieved using the numbers 3, 4, 6, and 8 exactly once.
To obtain 4: $$(8 - 6) \times 4 - 4 = 4$$ To obtain 10: $$6 + 4 = 10$$ To obtain 14: $$6 + 8 = 14$$

Number tricks You are given the number \( 3,4,6 \) and 8 . By using each number once exactly and possibly the addition, subtraction, multiplication and division can you obtain the number 4 ? The number 10 ? The number 14 ? You are allowed to use parenthesis if really needed.

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