Mind and Matter

Leo is reading Mind and Matter, a book written by a former NFL player now studying math at MIT. The book has 238 pages. If the pages are numbered consecutively starting from 1, what’s the sum of the digits of all page numbers? 

1+2+4=7

 In the expression I + II + IV = VII, I and V represent different single digit numbers. What’s I?

Hiding Red Paint

Leo puts 27 small 1x1x1 cubes together to build a 3x3x3 cube. He then paints all six faces of the big cube with red paint. After that, he rearranges the small cubes to a new 3x3x3 cube by minimizing the area of red paint on the surface. What is the red surface area of the rearranged 3x3x3 cube?

Pulling Weeds

Dr. Hong is taking Leo, Angela, and Elon to pull the weeds out of their front yard. If Dr. Hong does all the work on his own, it will take him 10 hours. If Leo and Angela work together, it will take them 15 hours. If Leo and Elon work together, it will take them 20 hours. If Angela and Elon work together, it will take them 30 hours. How many hours does it take if all four of them work together? Please keep two digits after the decimal point for the final answer. (We can assume that the weeds don’t grow when they work on the lawn.)

777 in Base k

 The three digit number 777 in base k is equivalent to 4145 in base 8. What is k?