Using only four 4s, make equations with answers 1 - 100

getmathhelp

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I have guessed my way through about 55 of these equations, using 4! 4 x 4 , 4 - 4 , 4 + 4, 4 divided by 4, the square root of 4, 4 squared, etc. However, I don't seem to be able to find any rhyme or reason for finding a formula to find the answers for the missing equations.

For example: 44 divided by 4 + 4 = 16,
44 - 4 - 4 = 36
4! + 44 divided by 4 = 35

I have found many (well 55 of the 100) equations, but really just by accident. I am hoping there is a formula.

Please help me :?
 


There is no formula for these types of exercises.

Educated trial-and-error, used with pattern recognition in some cases, is the way to go.

Journey onward! 8-)

If there's one that absolutely eludes all of your efforts, then post it, and we'll see if there are any hints for you.

 
Re:

mmm4444bot said:


There is no formula for these types of exercises.

Educated trial-and-error, used with pattern recognition in some cases, is the way to go.

Journey onward! 8-)

If there's one that absolutely eludes all of your efforts, then post it, and we'll see if there are any hints for you.


Correct. There's no formula. These problems are very open-ended, which is why I made a similar problem the subject of this competition
 
And even things like...

\(\displaystyle \frac{4^4}{4}=64\)

\(\displaystyle 4^{4-4}=1\)
 
Thank you for the encouragement. I have been working on the equations and have not had the time to solve all of them yet. But I am persevering. Thank you! :)
 
Denis, I don't know how you found the sites with the four 4's math puzzles on them. I searched and searched. Wish I had seen your post before i had to hand in my work....

Wish I could offer you a prize, but it was just an assignment. I don't even understand all the symbols used in completing some of the equations. For example, what is * :?:
 
getmathhelp said:
… what is *


On that page, it's used as a multiplication sign.

We use an asterisk to show multiplication because × looks too much like the letter x, which is used as a symbol for representing numbers.

In other words, 2x*x is clearer than 2x×x .

 
.
Well, there is a formula . . . if logs are allowed.

\(\displaystyle n \;=\;-\log_{\left(\frac{4}{\sqrt{4}}\right)}\left[ \log_4 \left(\sqrt{\sqrt{\sqrt{\hdots\sqrt{4}}} \right) \right]\)
. . . . . . . . . . . . . . . . .\(\displaystyle \backslash\_\_\_\_\_\_\_\_\_\_\_/\)
. . . . . . . . . . . . . . . . . n radicals

 
Denis said:
http://www.cut-the-knot.org/arithmetic/funny/4_4.shtml

Can I win your competition if I submit that, Doc? :D

Only if you're in grade 1 to 7, and send (optional) a nice postcard...

Oh, I might have to cross out the sqrts and !'s too, sorry.... :)
 
Using only factorials, sqrts, the floor function, and a single '4', it is possible to make every positive integer an infinite number of different ways.
 
getmathhelp!

I have guessed my way through about 55 of these equations,
. . using 4! 4 x 4 , 4 - 4 , 4 + 4, 4 divided by 4, the square root of 4, 4 squared, etc.
However, I don't seem to be able to find any rhyme or reason for finding a formula.

For example:
4[sup:3um0vrvz]4[/sup:3um0vrvz] divided by 4 + 4 = 16

44 - 4 - 4 = 36

4! + 44 divided by 4 = 35 ??

I have found many (well 55 of the 100) equations, but really just by accident.
I am hoping there is a formula.

If decimal points are allowed, a whole universe of possibilities exists.

\(\displaystyle \text{Note that: }\:\begin{Bmatrix}\dfrac{44}{.4} &=&11 \\ \\[-2mm] \dfrac{4!}{.4} &=& 60 \end{Bmatrix}\)

\(\displaystyle \text{So we have: }\;\begin{Bmatrix} \dfrac{4! + 4 - .4}{.4} &=& 69 \\ \\[-2mm] \dfrac{4! + 4.4}{.4} &=& 71 \\ \\[-2mm]\dfrac{\left(\frac{4}{.4}\right)!}{(4+4)!} &=& 90 \end{Bmatrix}\)


Many many years ago, I assigned this challenge to an advanced group of high school students.
I gave the challenge at 9 a.m. and by noon they had completed the list.
(The above examples were theirs.')

One student introduced repeating decimals.
\(\displaystyle \text{So that: }\:.\overline{4} \:=\:0.444\hdots \:=\:\frac{4}{9}\)
. . \(\displaystyle \text{and: }\;\frac{44}{.\overline{4}} \;=\;\frac{44}{\frac{4}{9}} \;=\;99\)

. . . Yike!

 
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