If five people are friends, how many friendships are there?

If five people are friends, how many friendships are there?

New year and perhaps new school, so “make new friends, keep the old” has a special intensity. You might build up to the title question, starting with, “If two people are friends, how many friendships are there?” 

Perhaps A’s friendship for B, B’s for A; A&B’s different friendships at school, at A’s house, at B’s house; their friendship during particular activities; their dormant friendship when they don’t see each other during a holiday.

With, “If three people are friends, how many friendships are there?” there are the two-way friendships and the friendship between all three simultaneously. But there’s also perhaps the difference between A’s friendship with B when C is there compared to when C is not, and so on. One to one friendships can be more intimate and group friendships more banterish, or one person might be at the centre of a friendship group that contains other existing friendships. 

You can’t, of course, answer the question without deciding what counts as a different friendship. Is a friendship defined by what you do, how you speak to one another, how you act towards one another, the customs or responsibilities of that friendship? This is at the overlap of two interesting areas of philosophy – eudaimonic philosophy (human flourishing, the good life) and ontology (the branch of metaphysics that looks at identity, what exists and so on).

By the time you get to the title question, you could be looking at 5x4x3x2x1 friendships or more, so for the mathematicians amongst you an opportunity too introduce factorials!

Training & Workshops

This week’s question emerged from a curriculum clinic session at Northcote School, last Monday. I’ve enjoyed being back in schools working face-to-face with teachers and students. Whether you are looking for an introduction, a refresh or to go deeper with your philosophy, get in touch. I have seven days still available for school visits this half term, which can be INSET, workshops with pupils followed by a twilight, or a Philosophy Festival in which I work with every child in your school in a series of assemblies and activities.

Online Sessions

Monday was the start of the new block of Zoom classes over at p4he.org As well as classes that are scheduled during the day for homeschoolers, we also have sessions in philosophy, debating, improv and now Shakespeare that are timed to suit all children, with some of the evening ones particularly suited to bright children who are a few years ahead of their peers. It’s always good for people to have extra friendship groups over and above their own school, and these groups are great places to meet and connect over lively, fun and challenging discussions. You can get a taster session for £7 or a block of six for £39. 

Best wishes,


P.S. We had the first free live training for Sticky Questions last week. Next one is Wednesday 22nd September and then Tuesday 28th, both at 4.30 p.m. The sessions are on Zoom and are free to schools that have bought the package for their schools. Get your PTA to invest in a set – about 3p per child per week!

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