Richard Williamson proved that all odd numbers are polite: This is because any odd number can be expressed as an even number + 1 and hence is the sum of half that even number, and the next integer. For instance, 13 = 6 + 7, 45 = 22 + 23, etc, where 6 and 22 are half of 12 and 44 respectively. As an odd number can always be expressed as the sum of two consecutive integers, and as this satisfies the definition of politeness, all odd numbers are polite.

Richard also found the consecutive sums for 544 and 434 and the way of characterizing impolite numbers, namely that they are all powers of 2. All other integers are polite. Shu Cao of the Oxford High School for Girls also proved this result.

James Newton of Hills Road Sixth Form College, Cambridge proved this result and also gave a nice generalization defining a 'degree of politeness' of numbers.