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## 4+4+4+4=2!

 Category: Math Submitted By: javaguru Fun: (2.42) Difficulty: (3.05)

In "4+4+4+4=1!" it was shown that just using addition, subtraction, multiplication, and division, you can form all the numbers from 0 to 9 using exactly four 4's.

For example: 7 = 4 + 4 - 4/4

Now if you add the operations square root, factorial, and exponent, what is the first number counting up from zero that can't be formed? How can each number be formed?

You may use parentheses, but you may not combine fours to make 44 or 444.

(An exponent is raising a number to the power of another number such as 4^4 = 256; a factorial is the product of all the integers from 1 to the integer such as 4! = 1x2x3x4 = 24.)

The first number that can't be formed is 39. This isn't proved, so maybe you can figure out how to form 39.

There are several ways to form most of the numbers, here are some for the numbers 0 to 9 using just addition, subtract, multiplication and division:

0 = 4+4-4-4 = (4-4)*4+4
1 = (4/4)*(4/4) = (4*4)/(4*4) = (4+4)/(4+4) = (4/4)/(4/4)
2 = (4/4)+(4/4) = (4*4)/(4+4) = 4/((4+4)/4)
3 = (4+4+4)/4 = (4*4-4)/4
4 = (4-4)*4+4
5 = (4*4+4)/4
6 = 4+((4+4)/4)
7 = (4+4)-4/4
8 = 4+4+4-4 = (4/4)*4+4 = ((4+4)/4)*4 = 4*4-4-4
9 = 4+4+4/4

Here are some ways to form the numbers from 10 to 38 using the additional operators:

10 = 4+4+4-SQRT(4)
11 = 4!/SQRT(4)-4/4
12 = 4*(4-4/4)
13 = 4!/SQRT(4)+4/4
14 = (4+SQRT(4))+SQRT(4)
15 = 4*4-4/4
16 = 4*4-4+4 = 4+4+4+4 = 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
21 = 4!-4+4/4
22 = 4!-(4+4)/4
23 = 4!-4^(4-4)
24 = 4*4+4+4
25 = (4+4/4)^SQRT(4) = 4!+4^(4-4) = (4!*4!+4!)/4!
26 = 4!+(4+4)/4
27 = 4!+4-4/4
28 = 4*(4+4)-4 = 4!+4+4-4
29 = 4!+4+4/4
30 = (4+4)*4-SQRT(4) = (4+4/4)!/4 = 4!+SQRT(4)+SQRT(4)+SQRT(4)
31 = ((4+SQRT(4))!+4!)/4! = (SQRT(SQRT(SQRT(4)))^4!-SQRT(4))/SQRT(4)
32 = 4*4+4*4 = SQRT(4)^(4+4/4) = 4^4/(4/SQRT(4)) = 4*4*4/SQRT(4) = (4+SQRT(4))^SQRT(4)-4
33 = (SQRT(SQRT(SQRT(4)))^4!+SQRT(4))/SQRT(4)
34 = 4*(4+4)+SQRT(4) = (4+SQRT(2))!/4!+4 = 4!+4+4+SQRT(4)
35 = 4! + (4! - SQRT(4)) / SQRT(4)
36 = 4! + 4 + 4 + 4
37 = 4! + (4! + SQRT(4)) / SQRT(4)
38 = 4! + (4 * 4) - SQRT(4)