Back in April, 2019, we wrote about the "Sum of Three Cubes" problem having been solved for the number **33**. The problem can be stated as:*k* = x^{3} + y^{3} + z^{3}, where *k* is a whole number.

For example,**29 = 3 ^{3} + 1^{3} + 1^{3}**

**26 = 114,844,3653 + 110,902,3013 + –142,254,8403**.

**RELATED: THE NUMBER 33 AS THE SUM OF THREE CUBES PROBLEM HAS JUST BEEN SOLVED**

Equations of this form, *k* = x^{3} + y^{3} + z^{3}, where *k* is between **1** and **100** are called Diophantine Equations, and are named after the Greek mathematician Diophantus of Alexandria who lived around **250 AD**.

### The 1950s yields solutions

Starting in the **1950s**, mathematicians began working on solving Diophantine equations, and they found solutions for all the numbers except **33** and **42**.

The numbers **4**, **5**, **13**, **14**, **22**, **23**, **31**, **32** can never be expressed as the sum of three cubes because they can be written as:

9 x *k* + 4 or

9 x *k* + 5, where *k* is any integer.

That's where things stood until **April 2019**, when Andrew Booker of the University of Bristol created a new computer algorithm to solve the sum of three cubes problem for the number **33** that examined the number line in both directions all the way up to 99 quadrillion. That's **99,000,000,000,000,000**.

The answer Booker found for **33** is:**8866128975287528 ^{3} + -8778405442862239^{3} + -27361114688070403**

But,

**42**remained unsolved.

### The mystery of 42

Lewis Carroll, author of the *Alice in Wonderland* books was actually a mathematician, and he was obsessed with the number **42**. There are **42** illustrations in *Alice's Adventures in Wonderland*. "Rule Forty-two" in that same work states, "All persons more than a mile high to leave the court," and in *The Hunting of the Snark,* the Baker has "forty-two boxes, all carefully packed, With his name painted clearly on each."

Fast forward **114** years, and English author Douglas Adams publishes his wildly popular book, *The Hitchhiker's Guide to the Galaxy*. In it, a supercomputer named Deep Thought spends **7.5 million years** pondering the ultimate question, unfortunately once it finds an answer, no one can remember what the question was. The answer, however, is "**42**".

So, what is it with the number **42**? First, it is the sum of the first **6** positive even numbers:**2 + 4 + 6 + 8 + 10 + 12 = 42**.

42 is a *pronic number*, which are numbers that are the product of two consecutive integers:*n*(*n* + 1)**2 x 3 x 7 = 42**.

42 is also an *abundant number*, also called an *excessive number*. These are numbers where the sum of their proper divisors is greater than the number itself. Divisors of **42** are:**1, 2, 3, 6, 7, 14, and 21**, which add up to **54**.

42 is also a *sphenic number*, which are positive integers that are the product of three distinct *prime numbers*. A prime number is a number greater than 1 that cannot be formed by multiplying two smaller numbers. In the case of **42**, it is the product of:**2 x 3 x 7 = 42**, where **2**, **3** and **7** are prime numbers.

### The magic of 42

You can create a *magic square* whose sum is **42**:

And, there is a magic cube whose vertices add up to **42**:

### The fall of 42

Andrew Booker knew he was going to have to go greater than **99 quadrillion** to find a solution to **42**, and he teamed up with Massachusetts Institute of Technology mathematician Andrew Sutherland, who hooked him up with Charity Engine. Charity Engine is a crowdsourced "worldwide computer" comprised of some **500,000** home computers around the world.

Charity Engine uses a computer's idle processing power, and it took **1 million** hours of processing time to solve the Diophantine Equation where *k* is equal to **42**. The answer is:**(-80538738812075974) ^{3} + (80435758145817515)^{3} + (12602123297335631)^{3} = 42**.

If you're thinking that the question of the meaning of life has been answered, ponder this: between the numbers **100** and **1,000**, the sum of three cubes problem hasn't been solved for the numbers:**114**, **165**, **390**, **579**, **627**, **633**, **732**, **906**, **921** and **975**.

Back to the drawing board.