BrainDen.com - Brain Teasers
• 0

## Question

In each expression:

Use exactly three 8's.

Use at most three square root symbols.

Use at most two factorial signs.

You can use some combination of addition, subtraction/negation, multiplication,

and/or division signs (+, -, *, /).

You can use parentheses.  One pair should be sufficient.

Concatenation is not allowed.

No other numbers, operations, or characters are allowed.

To start off, I am giving you this expression for a solution:

8 + 8 + 8

> > > Try to come up with as many as five other expressions for solutions. < < <

## Recommended Posts

• 0

Slight modification of the rules:

An expression isn't valid unless if at least one operation, or set of parentheses, or both

can be removed, and the expression is still equal to 24.

##### Share on other sites
• 0

I wish to clarify two things:

One, when you say "concatenation is not allowed", does that apply to the factorial signs? That is, does the rule eliminate the double factorial?

Two, is this rule modification correct?
"An expression isn't valid unless if at least one operation, or set of parentheses, or both

8‼ / (8 + 8) = 24

can be removed, and the expression is still equal to 24."
At a quick initial read I expected it was trying to eliminate solutions that are basically identical due to the communitive property of some mathematical operations, such as X+Y is basically the same as Y+X, and perhaps the associative property, such as X+(Y+Z) = (X+Y)+Z. Yet, a second and more thorough read it basically states that the solution must be enclosed within another solution or an additonal set of parentheses are required. This interpretation seems odd, thus the request for clarification.

##### Share on other sites
• 0

(√(√8×√8 + 8))!

(8 - √(8 + 8))!

(-(√(8 + 8) - 8))!

##### Share on other sites
• 0

Regarding post # 3 -- DejMar, concatenation refers to only making multi-digit numbers here.

It does not eliminate the double factorial sign.  Your solution is one of the correct ones.

I hadn't counted on that solution, so there are more than five other solutions.

Regarding post # 4, all are valid.

##### Share on other sites
• 0

Here is what I found.

F stands for factorial, S for square root.

I didn't use negation, since it doesn't add anything interesting.

Some parentheses are redundant, feel free to ignore them.

```(8+(8+8))
F(S((8+(S(8)*S(8)))))
F((8/S(S((8+8)))))
F((8/F(S(S((8+8))))))
F(S((8*S(S((8+8))))))
F(S((8*F(S(S((8+8)))))))
F((8-S((8+8))))
F(S((F(S((8+8)))-8)))
F(S((8+S((8*8)))))
```

##### Share on other sites
• 0

witzar,

first one -- I know you stated to ignore the ones with redundant parentheses.  However, all

parentheses have to be left off this one, because "8 + 8 + 8" manages on its own to equal 24.

Modified rule.

-------------------------------------------------------

fourth one -- The inside factorial sign would be left off, because the expression that is equal

to 2, is already equal to 2 factorial.  Modified rule. **

It's a variation/repeat of the third one.

------------------------------------------------------

sixth one --  ** See the immediate comment above.

It's a variation/repeat of the fifth one.

##### Share on other sites
• 0

A note regarding the rule modification. I see now the word that caused confusion was the word 'unless'.

##### Share on other sites
• 0

My computer isn't letting me put post # 8 in a quote box.

Let me leave out the word "unless" for my intended meaning.

"Slight modification of the rules:

An expression isn't valid if at least one operation, or set of parentheses,

or both can be removed, and the expression is still equal to 24."

##### Share on other sites
• 0

8×√(!√(8+8)) = 24

[Here the factorial sign preceding the number is the common notation for the subfactorial function. The subfactorial (derangement) of 4 is 9.]

##### Share on other sites
• 0

DejMar,

this is getting away from me.  I intended the factorial symbol to just be used as a factorial, not also

as a subfactorial.  Also, on another site, a user stated that the double factorial is one character,

so I don't know if that should still be considered.

##### Share on other sites
• 0

Perhaps check it again,

It was my suspicion that you did not expect the factorial character to be used in the manner of a subfactorial. I also guessed you were not aware of the multifactorial notation a!(b), sometimes written a!(b), a!b, or a!b such that a - where a mod b = 0, and b +, but more often recognized with b (i.e., multiple) factorial characters following the number. Of course it would be difficult to find a solution for the multifactorial given the limitation of three 8's and the other notation symbols.
There indeed is a special character (i.e., a single code point) for the double factorial character -- just as I had used in my post for the double factorial -- hence my question about concatenation with the symbol. Of course, two factorial characters are often used in ASCII notation, as the code point is extended out of the normal ASCII range.

Edited by DejMar

## Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account. ×   Pasted as rich text.   Paste as plain text instead

Only 75 emoji are allowed.

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

×