# The 6s Math Puzzle

This is a pure mathematical puzzle which evaluate your ability to solve maths equation. This is a great problem for building number sense.

## Puzzle

The challenge is to make each below equation true using common mathematical operations. You cannot introduce any new digits (so the cube root ∛ is not allowed since it has a 3).

**Hint:** common mathematical operations are `+`

`-`

`x`

`/`

`!`

`√`

```
0 0 0 = 6
1 1 1 = 6
2 2 2 = 6
3 3 3 = 6
4 4 4 = 6
5 5 5 = 6
6 6 6 = 6
7 7 7 = 6
8 8 8 = 6
9 9 9 = 6
```

## Solution

**Note:** Please try on your own before looking at the solution. Also some of the equations has multiple solutions.

Let’s start with some easier ones.

```
2 + 2 + 2 = 6
6 + 6 – 6 = 6
6 × (6/6) = 6
7 – 7/7 = 6
5 + 5/5 = 6
```

Here are a few solutions for 3.

```
3 × 3 – 3 = 6
3! + 3 – 3 = 6
3! × (3/3) = 6
√(3 × 3) + 3 = 6
```

For the number 9, we can use a trick. Since √(9) = 3, we can take the square root of each number, so the problem is equivalent to solving 3 3 3 = 6, which we have just solved! So we can use any of those solutions, or we can find others too.

```
√9 × √9 – √9 = 6
(√9)! + √9 – √9 = 6
(√9)! × √9/√9 = 6
(√9 × √9/√9)! = 6
```

We can do a similar trick for 4. Since √4 = 2, we can use the solution for 2 2 2 = 6. But there are other solutions too.

```
√4 + √4 + √4 = 6
(4 – 4/4)! = 6
(√4 + 4/4)! = 6
```

Now we just have a couple more to solve and we will use 3! = 6 in many of the answers. We can solve for 10 as:

```
(√(10 – 10/10))! = 6
```

Then we have the 1 solution:

```
(1 + 1 + 1)! = 6
```

To solve 0, we use the fact that 0! = 1, and then we have reduced the problem to solving 1 1 1 = 6, which was previously solved.

```
(0! + 0! + 0!)! = 6
```

We just have one more to solve, which many people consider to be the hardest. One way to solve uses nested square roots.

```
8 – √(√(8 + 8)) = 6
```

The other method uses 3! = 6.

```
(√(8 + 8/8))! = 6
```

And we are done! Here are the above solutions listed in numerical order. This is not a comprehensive list. you might have found another way too!

```
(0! + 0! + 0!)! = 6
(1 + 1 + 1)! = 6
2 + 2 + 2 = 6
3 × 3 – 3 = 6
3! + 3 – 3 = 6
3! × (3/3) = 6
√(3 × 3) + 3 = 6
√4 + √4 + √4 = 6
(4 – 4/4)! = 6
(√4 + 4/4)! = 6
5 + 5/5 = 6
6 + 6 – 6 = 6
6 ×(6/6) = 6
7 – 7/7 = 6
8 – √(√(8 + 8)) = 6
(√(8 + 8/8))! = 6
√9 × √9 – √9 = 6
(√9)! + √9 – √9 = 6
(√9)! × √9/√9 = 6
(√9 × √9/√9)! = 6
(√(10 – 10/10))! = 6
```