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Number Theory

Number Theory is number magic. It's the study of how math itself is structured. And it deals with the coolest numbers: primes!

Secret Messages


On the first day of medical school, a professor writes her name on the board in a secret code:

\[\huge\text{ES. TNJUI}\]

What is the actual last letter of her last name?

Additional Information:

The code used is called a "Caesar Shift." Pictured below is a "decoder ring" used to encode and decode messages into and out of this code and a family of similar codes. In real life, the inner wheel of letters would spin independently of the outer wheel of letters.

Number theory plays a key role in encryption--the process of hiding and encoding information. Encryption enables the military to pass on messages secretly, merchants to pass credit card numbers to banks, and everyone to sign in (passwords) and check encrypted emails.

Numerical algorithms are one of the key ways engineers encrypt information. Algorithms turn messages into code that is impossible to read until the message is decoded.

The pigpen cipher replaces letters for symbols based on the grid below.

For example, the message "WORDS" encoded with the pigpen cipher is shown below.

What is the secret message encoded below?

A magic word is needed to open a certain box. A secret code assigns each letter of the alphabet to a unique number. The code for the magic word is \[9 \ 23 \ 14\ 14\ 18\ 5.\] What could be the magic word?

In a certain code language, three messages are encoded as follows:

  • Cats don't like apples \(\Rightarrow\) pic sip dip rit
  • Parrots really like apples \(\Rightarrow\) dip pic tif nit
  • Cats like parrots \(\Rightarrow\) sip dip tif

How is the word "don't" written in the code language?

To send a private message, Alan encodes it by rearranging the letters.

To do so, he first writes the word across two rows, switching between the top and bottom row with each later, as in the sample below:

He then reads the message across each row, starting with the top one and then moving to the bottom. In this case, the word "SAMPLE" would become "SMLAPE" once it is encoded.

Alan sends the message "ATRNLEIG." What was his original word?


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