# Fix These Sixes!

[Image courtesy of Daily Brain Teaser.]

It’s a new year, and I’ve already got a new math puzzle for you. A friend discovered this one and sent it my way in the hopes that I’d be able to crack this diabolical brain teaser.

Fellow puzzlers and PuzzleNationers, this is the Sixes puzzle.

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

The goal is to make all of the above equations true by adding mathematical symbols.

And, naturally, there are rules.

• You can use as many mathematical symbols as you want for each equation
• You are not allowed to use letters, so spelling out functions like “cos” is out
• You are not allowed to add digits of any kind, so no turning a “2” into a “12”
• The result has to be exactly 6 ( not 6.0000000000000001 or 5.999999999999999 )
• Square roots are allowed
• You are not allowed to change “=” to “≠” (not equal to) or manipulate the result in any way

Take a crack at it, then scroll down past the dice to see how it’s done.

[Image courtesy of The Progzilla Files.]

Let’s start by knocking out the easiest one.

2 2 2 = 6 can be resolved as 2+2+2=6.

6 6 6 = 6 can be resolved as 6+6-6=6 or 6-6+6=6.

They get a little trickier from here, involving multiple operations.

3 3 3 = 6 can be resolved as 3×3-3=6, becoming 9-3=6.

Fractions also come into play for a few of these equations. (This can also be represented as division.)

5 5 5 = 6 can be resolved as 5+(5/5)=6, becoming 5+1=6.

7 7 7 = 6 can be resolved as 7-(7/7)=6, becoming 7-1=6.

[Image courtesy of TAF.org.]

Okay, we’re halfway there, and now the square root rule gives us a hint regarding how to resolve the 4 equation.

4 4 4 = 6 can be 4+4-(√4)=6, becoming 8-2=6.

And this formula gives us a way to crack the 8 equation.

8 8 8 = 6 can be resolved as 8-√(√(8+8))=6, becoming 8-√(√16)=6, becoming 8-(√4)=6, becoming 8-2=6.

Square roots also come into play in solving the 9 equation.

9 9 9 = 6 can be resolved as (9+9)/(√9)=6, becoming 18/3=6.

Now, admittedly, at this point, I was stumped. I had two equations left, and no ideas regarding how to proceed.

0 0 0 = 6

1 1 1 = 6

So, I reached out to a mathematician pal of mine — the same one who helped me crack the diabolical Seesaw Puzzle from Brooklyn Nine-Nine — and he immediately knew what to do: use an exclamation point.

In mathematics, an exclamation point represents a factorial, the product of every positive number between the given number and zero.

For instance, 6! represents 6x5x4x3x2x1, or 720.

1 1 1 = 6 can now be resolved as (1+1+1)!=6, becoming 3!=6, becoming 3x2x1=6.

But what about 0 0 0 = 6?

Factorials to the rescue again! You see, 0! equals 1. So we can use the 1 equation as a template for this one.

0 0 0 = 6 can be resolved as (0!+0!+0!)!=6, becoming (1+1+1)!=6, becoming 3!=6, becoming 3x2x1=6.

And there you have it, the Sixes puzzle conquered with nothing but crafty math and puzzly skills. An excellent start to a new year of brain teaser challenges!

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