Can you make all the numbers from 1 to 100 using exactly four fours for each number? Some examples: (4+4)/(4+4) = 1. 4/sqrt(4) x 4/4 = 2.

This is a very old problem. I won't get into the history here, but if you are interested, an internet search will yield many pages on the topic, and many pages of various solutions. As I was working on the solution, I got stuck on 73. I mentioned it to Dan F at work, and he came up with a solution for 73. And as Dan does, he pointed out a way of approaching it that I hadn't considered which led to a revision of my solution in which I tried to find solutions in what I call families. The biggest family is the 4!/F ± T family. 4! is 4 factorial which equals 4 x 3 x 2 x 1 = 24. F represents a number that can be represented using one Four. There are 6 of them as far as I know. 24, 4, 2, 2/3, 4/9, 4/10. 2 = the square root of 4. 2/3 = the square root of point 4 bar. Point 4 bar is like .4 with a bar over the 4. It means the 4 repeats endlessly and the decimal .4444... equals 4/9. 4/10 is just .4. Getting back to the formula, T represents the whole numbers from 0 to 96 which can be represented using two fours. So 4!/F ± T uses 4 fours. And 79 of the numbers from 1 to 100 can be made using this formula. The next biggest family of numbers I found was the 4!F ± T. It has 11 members. So between the 2 families, 4! times or divided by F plus or minus T, we can represent 90 of the 100 numbers.

I've posted my solution of the Four Fours Problem in a Google Spreadsheet at

Four Fours.

All hundred numbers contain a 4! in them. Most of the numbers in T are simple to generate, but I want to explain how I get 32 using two fours. Before reading further you may want to see how many whole numbers you can come up with from 0 to 100 using two fours and see if 32 is in your list. OK, here is the explanation. Do you know that the cube root of 8 equals 2, because 2x2x2=8? And the 4th root of 81 equals 3, because 3x3x3x3=81? Well the point 4 th root of 4 equals 32. So using two 4's and a radical sign you can get 32 with two fours. If you come up with any numbers using two fours that I don't have listed in T I'd like to know. Maybe then I could fit the 10 remaining numbers into one of the two big families.