You are to find expressions for each whole number from 1 through 20, using in each expression exactly four 4�s, plus any mathematical symbols you like: parentheses, plus signs, etc.
Sample: 
1 = 
4 + 4 
. 
4 + 4 
Show/Hide Solution
There are many possibilities. Here are some:


11 = 4 + 4 + 4 – [√√4]








13 = 4 + 4 + 4 + [√√4]


4 = √4 + 4 + 4 + 4









  






8 = √4 + √4 + √4 + √4 





19 = 4 · 4 + √4 + [√√4]





Note: [] is the greatest integer function. That is, [x] is the greatest integer less than or equal to x.
For example, [√√4] = [√2] ≈ [1.4] = 1.