Represent 6 with three Identical Digits and Arithmetic Operations

I was just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an extension of the following arithmetical formula: 2 + 2 = 4. Every philosophical proposition has the more general character of the expression a + b = c. We are mere operatives, empirics, and egotists until we learn to think in letters instead of figures.

Oliver Wendell HOLMES

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6 is such a number which can be obtained using any number from 0 to 9 three times, with the help of some arithmetical operators. In this blogpost, our aim is to make the following equations correct by using mathematical operations.

1. 1  1  1 = 6
2. 2  2  2 = 6
3. 3  3  3 = 6
4. 4  4  4 = 6
5. 5  5  5 = 6
6. 6  6  6 = 6
7. 7  7  7 = 6
8. 8  8  8 = 6
9. 9  9  9 = 6

Here, arithmetical operations include addition, subtraction, multiplication, division, square root, cube root, factorial, fraction, exponentiation etc. Occasionally, integer floor function is used.

We know that 0! = 1 and 3! = 6. Therefore,

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

Since cos (π/2) = 0, another representation may be

6 = [{cos (π/2)}! + {cos (π/2)}! + {cos (π/2)}!]!

Since sin (0) = 0, another representation may be

6 = [{sin (0)}! + {sin (0)}! + {sin (0)}!]!

Since 3! = 6, so

6 = (1 + 1 + 1)!

Since x⁰ = 1, another representation may be

6 = [x⁰ + x⁰ + x⁰]!

Since cos (0) = 1, another representation may be

6 = [{cos (0)}! + {cos (0)}! + {cos (0)}!]!

Since sin (π/2) = 1, another representation may be

6 = [{sin (π/2)}! + {sin (π/2)}! + {sin (π/2)}!]!

Normal representation is

6 = 2 + 2 + 2

Since 2! = 2, another representation may be

6 = 2! + 2! + 2!

Few representations are given below.

Here, we use square root concept. Few representations are given below.

Few representations are given below.

Few representations are given below.

Few representations are given below.

Integer floor function (or greatest integer function) gives us the largest integer less than or equal to the given number. Few representations are given below.

Few representations are given below.

Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call “Math1089 – Mathematics for All!“.