Nibcode Solutions Blog

There are several logical grid puzzles you can find on the internet, but undoubtedly, one of the most popular known is the Einstein's riddle. Although it's attributed to Einstein, there's no evidence to back up that claim. It's rumoured that only 2% of the world can solve it, but it's really not so hard, all one needs is logic and deduction. In this article we are going to present a method based entirely on matrix operations to solve this riddle and any other similar.

The riddle

There are five houses in five different colours in a row. In each house lives a person with a different nationality. The five owners drink a certain type of beverage, smoke a certain brand of cigar and keep a certain pet. No owners have the same pet, smoke the same brand of cigar, or drink the same beverage. The question is: Who owns the fish?

Nowadays information is at hand. Just open your browser, type some words about what you want to know, and thousands of pages will arise to satisfy your information needs; and surprisingly, you will find the most relevant information in the first proposed pages. It’s not a mystery the way the search engines work to make their magic, it’s just mathematics. Behind their complex algorithms is hidden the linear algebra and other mathematical theories.

Every search engine needs three basic elements: a web crawler, a database to save the data it finds, and an algorithm to determine the order of pages returned by any search inquiry. The two first elements can be automated easily, the main problem lies on the third one, and here is where the mathematical tools are the key to the solution.

Since humans invented the written language, they have tried to share information secretly. This is basically, the objective of Cryptography, the study of the techniques to protect sensitive communications by means of data encryption and its posterior decryption. Encryption is the transformation of data into some unreadable form, so, even those who can see the encrypted data cannot understand the hidden information. Decryption is the reverse of encryption; it is the transformation of encrypted data back into some intelligible form.

Although there are different methods to encrypt and decrypt messages, we'll focus on a linear algebra based cipher, the Hill cipher, which uses a matrix as a cipher to encode a message, and it's extremely difficult to break when a large matrix is used. The receiver of the message decodes it using the inverse of the matrix.

Many people find in the lottery the chance to change their life, but the reality is that the odds are so low that most people can be playing for hundreds of years without ever get closer to the prize. A thought usually goes through the mind of the players is: "If there are people who won the price, why cannot I win it too?", which I do not doubt, but let us look at the numbers and see how lucky we would be if we won the lottery.

Let's explain how to calculate the chance to win the lottery based on 49 possible numbers, where 6 numbers are randomly selected in each drawn.