Maths: The game of Srinivasa

This is a partly solved problem, and I used to give it to my brighter students under the name Srinivasa, after this chap. It was a game I invented, so I get the naming rights! (As a botanist, working in a sanctuary, when people ask me the name of a plant I don't know, I always answer 'Albert', so you are lucky to get Srinivasa!)

Some of my students were in a maths competition where they had to generate as many integers as they could from the year date for that year. They had to use all of the four digits in any order, and after that was over, I challenged them to play the game, but now using 1729 as their starter.

There are many ways that you can manipulate a group of numbers to obtain give different values. Your task is to find solutions using the numbers 1, 7, 2 and 9 in that order, once only, combined by any mathematical symbols such as exponentiation, square roots, multiplication, addition, subtraction and division, with as many brackets as you like. You can even use factorial notation (5! = 5 x 4 x 3 x 2 x 1) or any other notation, but no other numbers.

For example, 1 = 1⁷²⁹ = -17 + (2x9) = 1 -7 -2 +9 etc.;

or 6 = 17 – 2 – 9;

or 7 = -1 + 72/9;

or 8 = (1 x 72)/9;

or 9 = 1 + 72/9;

or 10 = 1⁷² + 9 = 17 + 2 – 9;

or 12 = -17 + 29 = (1 + 7)/2 x √9;

or 30 = 1⁷ + 29;

or 36 = 1 x 7 + 29;

or 46 = 17 + 29.

When I posted this on the web site that was the beginning of a whole book, and now this blog, I had lots of help from Klaus Marx and Frank Schnell, then at Robert Bosch Gmbh in Germany, who helped me find solutions for every integer up to 50. Frank was still at Bosch in 2018, but Klaus had retired to Northern Italy, not that far from where Leonardo Pisano Bigollo was born. Like me, they still play with number things, so watch out for them around the traps. But who was LeonardoPisano Bigollo? I will go there next, just after this: 

Curiosity of no importance: 729 is 9³, and it is also the sum of three cubes: 1³ + 6³ + 8³, but as 6³ = 3³ + 4³ +5³, 729 is also the sum of five cubes: 1³ + 3³ + 4³ +5³ + 8³.

And another oddity:

Something I came up with during editing: can you come up with a 1, 7, 2, 9 sum that returns the value 91? I can get to 82, but that’s all…

I’m not saying how I did it, but of the first 2000 integers, 150 of them can be expressed as the sum (or difference) of two cubes. I will admit to using both a spreadsheet and a word processor with a SORT function. That's all: Do It Yourself!

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