# Math Circle Session 4: Ciphers

Today saw the fourth session of our math circle. Attendance was down, and there was a sombre mood, due to a tragic event to hit the the school community today.

We went ahead with the three children who came, looking at ciphers. Following the guidance and resources in Rozhkovskaya’s book, we looked at the following activities:

• We started with some plaintext words and one ciphertext word, and the kids very quickly spotted, without any help, which plaintext word corresponded to the ciphertext word, on the basis that both had a double letter in a certain place and they had the same number of symbols.
• We showed how a coding wheel can be used to produce a Caesar shift cipher, and got them to actually build such a wheel.
• We queried how many possible ciphers could be constructed with such a wheel. Interestingly, this was not obvious to the children at first, and a common answer was “infinite”. On questioning, this is because they were considering all integer shifts, i.e. mapping the letter $n$ to the letter $n+k$ for some key $k$, and forgetting the modulo arithmetic in play, despite having the wheel in front of them.
• We got them to produce coded messages to each other using the wheel, which they seemed to enjoy. Our initial plan was to have them pass the message and the key to the next child to decode. However, they immediately attempted to decode without they key by looking for patterns, so we didn’t stop this. One key pattern they spotted included the limited number of two letter words in English, allowing them to enumerate possible keys on two-letter words in the ciphertext. Another was the limited number of letters in the alphabet that can appear twice consecutively in an English word, again allowing for enumeration.

There’s a lot more that can be done with ciphers, and I expect we’ll continue this for another session.