Classification of the limitation in the choice of math-symbols

  We are sure the following assessments.

• Odd numbers would more difficult be expressed by four "4"s than even numbers since "4" is an even number.
• The larger number would be the more difficult to be expressed by four "4"s than the smaller numbers (suppose if 1000000 could be calculated by four "4"s.)
• The four fundamental rules of arithmetic must be more valuable than alphabetic functions (or operators)(like , , and etc.)
• Non-alphabetic(operators) functions are considered more valuable than alphabetic functions (operators).
• , e or i can be used, but since it is possible that they are numbers other than "4", these should get least marks.

According to the above assessments, we define the way how to measure the difficulty.

The operations marked by "*" get penalty which equals to the 8-folds number of appearance. Therefore if you use "+" twice, you will get . If you use e, you get 0 because the mark gets one penalty. But as an exception, even if you use e twice, you will not get two penalties, because we assume the make must not be negative in each assessment.