# 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.

``````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
``````