Whilst Tuesday the 12th of January 2021 feels pretty ordinary, if at some point you jot down the shorter format ‘12/1/21' you will notice today is, in fact, a palindrome day!

Palindromes are sequences of figures or letters that appear the same when written forwards or backwards, a bit like the concept of symmetry in shapes but with words and numbers. For example the name ‘Hannah’ is palindromic, as is the number 123321 and the sentence “Was it a car or a cat I saw”. Palindrome enthusiasts (a group that tend to overlap somewhat with mathematicians) are particularly satisfied whenever the date shows this property and have coined the term ‘palindrome day’ to describe it.

To mark the occasion, here are 5 interesting palindrome facts both mathematical and not so much.

1 - A FORMAL DEFINITION

Okay so not quite a fun fact but in maths a formal definition is usually a good place to start.

Consider some number n>0 written in the standard notation in base 10 with k+1 digits aas

n = i=0kai10i

with, as usual, 0 ≤ ai < 10 for all i and ak ≠ 0. Then n is palindromic if and only if ai = aki for all i. This definition can be applied more generally to any base ‘b’ by substituting that ‘b’ for 10.

2 - SAIPPUAKIVIKAUPPIAS

This 19 letter finnish word for a dealer in lye is officially recognised as the longest palindromic word, it’s even in Guinness World Records.

3 - PALINDROMIC PRIMES

Some prime numbers are also palindromic, for example 11, 191, 727 and the largest known base 10 palindromic prime 10474500 + 999 × 10237249 + 1 which was discovered by Serge Batalov in 2014 and has over 474,501 digits.

4 - BINARY PALINDROMES

Changing the base of a number may also change whether it is palindromic! Take the decimal number 107, clearly not palindromic. When it is instead expressed in it’s binary form however it is written as ‘1101011’ and so is then a palindrome! The sequence of binary palindromic primes is particular interesting for mathematicians as both the Mersenne and Fermat primes are contained within it!

5 - EMORDNILAP

Clearly the word ‘palindrome’ is not palindrome however that hasn’t stopped it being flipped to create a whole new word “emordnilap’ (don’t worry we’d never heard of it either..). It is used to describe words that when spelled backwards are another word, for example ‘dog’ and ‘god’ or ‘drawer’ and ‘reward’. Whilst neither a ‘palindrome’ or ‘emordnilap’ are palindromes, if emordnilap makes it into the Oxford dictionary any time soon then both words will become emordnilaps!

- Lucy Chats Maths