This is the story of a Chinese Emperor and a farmer. “The emperor asks the farmer to ask how much payment he wanted. It is said that the farmer asked for 1 grain of wheat for the first square, 2 for the next, 4 for the third, and 8 for the fourth, etc. for the 64 squares on the chessboard. How much wheat did he receive?”
You can read the text in this Google ebook and try out the Excel formulas in the attached file (Mod function used heavily). The sum of all the grains would be 2^64-1. You can download the excel file here to know the formulas better.
Another property of the power series of two is that there seems to a trend in the number of occurrences of the first digit. Eventually it came to be known as Benford’s law. After watching this Numberphile video, using the Left function in Excel, I extracted the first digit, converted the text to a number by adding zero and then counted the frequency of digits from 1-9 and it fits very well to log(1+1/digit). I have made use of Indirect function to count the digits from the first 100, 200, 300 rows and so on.
Like my previous post on Π, you can again use conditioning formatting to make the artwork of your choice using the char function. Playing with numbers is indeed very interesting. Thanks to Microsoft Excel, everyone has the power to analyse the numbers. For example, does bases other than two obey Benford’s law ? Could we find out how far the first number if from ten and try to construct a new number ? You can obtain the excel file I used here.
When Ramanujan was asked how he got the formula for infinity, he said that God speaks to him. Bear in mind he was just an ordinary clerk. It is befitting that Hollywood has made a movie on Ramanujan, the man who knew infinity.