The Four 4's Game

[Crossbreed]

(I think that 31 & 33 are going to be the next hard ones.)

I found out that there were more functions allowed in this "game."

Use exactly four 4's to form every integer from 0 to 50, using only the operators +, -, x, /, () (brackets), . (decimal point), x² (square), square root and ! (factorial).

Example: 0 = 44-44

Wikipedia includes the ^ (exponentiation) function.

 

[Nervous Rex]

28 = √4 * 4^√4 - 4

I had to use ^ for “to the power of”. @Crossbreed, how did you make get superscript?

 

[Nervous Rex]

I found out that there were more functions allowed in this "game."

Yes, absolutely! Go to town on creativity and new functions. I took the rules to mean that you can write anything you want that makes math, but you cannot write any digit except 4, and you must use exactly four of those.

I worked out that I can hit 31 if i can use the Floor and Ceiling functions.

Floor rounds down to the nearest integer: [[3.5]] = 3, or [[√√4]] = [[√2]] = [[1.414]] = 1
Ceiling rounds up: ]]3.5[[ = 4, or ]]√√4[[ = ]]√2[[ = ]]1.414[[ = 2
There are other notations. You could just as easily write Floor(√√4).

 

[Crossbreed]

...how did you make get superscript?

I just Googled [unicode superscript] and copy/pasted from the Wikipedia article.

 

[Crossbreed]

I worked out that I can hit 31 if i can use the Floor and Ceiling functions.

I saw two more elegant solutions for 31 & 33 (of which I recuse myself*).

31 was accomplished by the methods we have used up until now.
33 made use of .4 .

I hadn't considered using .4, 4.4 & 44 before reading that. Also promising, .4 (with an overscore for a repeating decimal). It is equal to 4/9ths. Can we agree to the notation, ".4~" ? (See my next post.)

*Or I will submit a different solution.

 

[Jeremiah]

29 = 4! + 4 + 4 / 4

 

[Crossbreed]

Also promising, .4 (with an overscore for a repeating decimal). It is equal to 4/9ths. Can we agree to the notation, ".4~" ?

I found a computer-friendly notation, .(4) .

 

[Crossbreed]

In studying !4, I just learned about subfactorials & derangement.

Yay! me.

Factorial counts the number of permutations for four different people to sit in four different chairs.
Subfactorial/derangement counts the number of ways for four different people to sit in four different chairs without any of them repeating their first chair.

!4 = 9

 

[Nervous Rex]

In studying !4, I just learned about subfactorials & derangement.

Yay! me.

That's pretty cool. Maybe I should edit the first post in this thread and add a list of all operations used so far.

I can get to 31 with !4, or with .(4). I haven't figured out a way to do it without those.

 

[Crossbreed]

Maybe I should edit the first post in this thread and add a list of all operations used so far.

Or you could link to the Wikipedia article.

 

[Crossbreed]

30 = 4 x 4 x √4 - √4

 

[Nervous Rex]

Okay, let's get past 31. I came up with:

31 = 4! - √4 + 4/.(4) ( .(4) = 4/9, so 4/.(4) = 9)
31 = 4! - 4/√4 + !4 (!4 = 9, as shown above)
31 = 4! + 4 + √4 + [[√√4]] ([[x]] is floor of x, or x rounded down to the nearest integer. I think this solution is just plain ugly)

I did see the solution that uses only +-*/! on the Wikipedia article. I wish I had found that solution on my own.

 

[Crossbreed]

[[x]] is floor of x, or x rounded down to the nearest integer.

Outside of a computer, does "floor" mean "lowest integer" or "integer closest to zero?"

Does [[-5.5]] = -6 or -5?

 

[Nervous Rex]

Outside of a computer, does "floor" mean "lowest integer" or "integer closest to zero?"

Does [[-5.5]] = -6 or -5?

What do you want it to mean? <Holds hand out for bribe>

Calculators disagree on it.

The math standard says that floor always goes lower. Since -6 < -5.5, goes [[-5.5]] = -6. On a number line that counts up to the right and down to the left (etc., -3 -2 -1 0 1 2 3, etc.), floor always goes left and ceiling always goes right. Shall we stick to that definition?

 

[Crossbreed]

Shall we stick to that definition?

Sure, I just wasn't certain what the standard was. (VBASIC added Fix() for this very reason. Int() acts more like Floor().)

 

[Jeremiah]

32 = 4 * 4 + 4 * 4

 

[Crossbreed]

33 = 4! + !4 + 4 - 4

Here is the clever one that I read,
33 = (4 - .4)/.4 + 4!

 

[Nervous Rex]

34 = 4! + 4 + 4 + √4

Here is the clever one that I read,
33 = (4 - .4)/.4 + 4!

That is pretty cool. I read this one for 31:
31 = 4! + (4! + 4)/4

 

[Nervous Rex]

35 = 4! + !4 + 4/√4
36 = !4 + !4 + !4 + !4

Also, @Crossbreed, you sent me on a major diversion, figuring out how to calculate subfactorials for larger numbers. Fun stuff!

 

[Crossbreed]

37 = !4 x 4 + 4/4