### Four 3's

Using arithmetic combinations of four 3's express the integers from 1 to 15.

You must use all four 3's for each integer.

You may combine the 3's using the arithmetic operations +, -, * (multiplication) and / (division).

You are also allowed to use sqrt (square root) (e.g. sqrt(3) ).

You are allowed to use concatenation (e.g. 33 is valid as part of an answer).

You are allowed to use factorials. For example, 3! (or factorial(3)) = 3*2*1 = 6. (The factorial of an integer x! is the product of all the positive integers less than or equal to x).

You do not need to use anything other than those listed above - so no decimal places, powers of 3, trig operations, logarithms, cube/quartic/etc. roots, etc.

## 4 Comments:

1 = 33/33

2 = (3!)/((3*3)/3)

3 = (3*(sqrt(3)*sqrt(3)))/3

4 = ((3*3)+3)/3

5 = (3+3) - (3/3)

6 = 3+3+3-3

7 = (3+3) + (3/3)

8 = (3! + 3) - (3/3)

9 = 3 + 3 + (3^2/3)

10 = (3! + 3) + (3/3)

11 = 3! + 3! - (3/3)

12 = 3 + 3 + 3 + 3

13 = 3! + 3! + (3/3)

14 = (33/3) + 3

15 = (3*3) + 3 + 3

All ok except you don't need to use the 2 in 3^2 for 9....

9 = 3 + 3 + (sqrt(3)*sqrt(3) ) works!

or 9 = (3 * 3) + (3 - 3) ...

just do (3! + 3)+3-3

what does sqrt mean?

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