The Four Fours Problem

[James Chegwidden]

Here is a problem that I found in a rec math book from the 1950's
This is an old problem consisting of expressing successive integers (to a limit) in appropriate mathematical form, using only 4 (four) fours (4's) in each expression together with any necessary signs.
Note: My math collegues had a field day with this problem.
Examples: 1 = 4/4 * 4/4
10 = (44 -4)/4
So for what are the first twenty numbers written in only 4 4's.

[Melvin Menezes]

1 ==44/44
2 ==(4*4)/(4+4)
3 ==(4+4+4)/4
4 ==4*(4^^(4-4))^^ is power
5 ==4+(4^^(4-4))^^ is power
6 ==4+((4+4)/4)
7 ==4+4-(4/4)
8 ==4+4+4-4
9 ==4+4+(4/4)
10 ==(44-4)/4
11 ==
12 ==
13 ==
14 ==
15 ==44/4 +4
16 ==4*4*4/4
17 ==
18 ==
19 ==
20 ==(4+(4/4))*4

[Eric Pascarello]

12=(44+4)/4

[David Willis]

11 == (4/.4) + (4/4)

[Greg Harris]

this is assuming i can use mod(x,y) and !
1 = (4/4)*(4/4)
2 = (4*4)/(4+4)
3 = ((4*4)-4)/4
4 = (4*(4-4))+4
5 = MOD((4/4),4)+4
6 = (SQRT(4))+(4-4)+4
7 = ((4+SQRT(4))+(4/4))
8 = ((4+4)*4)/4
9 = (4+4)+(4/4)
10 = (4!+SQRT(4)-4*4)
11 = (4!/SQRT(4))-(4/4)
12 = (4!+4-4*4)
13 = (4!/SQRT(4))+(4/4)
14 = (4*4)-(4/SQRT(4))
15 = (4*4)-(4/4)
16 = (4/4)*(4*4)
17 = (4*4)+(4/4)
18 = ((4*4)+4)-(SQRT(4))
19 = (4!-4-4/4)
20 = (4*(4/4+4))

[Melvin Menezes]

17 == (4*4)+(4/4)
Ah! Greg hit post button before i did!
Using a . was smart, David.
[ June 03, 2003: Message edited by: Melvin Menezes ]

[Greg Harris]

sorry... i should have posted a link to my attempt, rather than the solutions.
i wish i had thought of 44 and 0.4... things would have been much easier!
(oops, i used html code)
[ June 03, 2003: Message edited by: Greg Harris ]

[Mark Herschberg]

I did a variation of this in 6th grade. You can do all the numbers 0 to 100.
--Mark

[Greg Harris]

actually, after googling around, this guy says you can go from 0 to 40,000.
here is a short list that goes from 0 to 1,000.
not quite the same as what i came up with... and i did not think about using sin, cos, gamma, etc.

[John Lee]

there is an old game: pick any 4 playing cards, use only "+", "-", "x", "/" to get 24.
the introduction of "mod", "sqrt", and so on makes it simpler. so why i can not use self defined functions?
[ June 04, 2003: Message edited by: Don Liu ]

[Greg Harris]

> so why i can not use self defined functions?
well, if you provide a proof with your function, then you can use it!

[John Lee]

let CVFD(4444) = 21,
so,
21 = CVFD(4444)
for any number X, i can define a function XXX(), let XXX(4444) = X

[Greg Harris]

nice