Can you explain a paradox to a seven year old?

During the current lock down in the UK schools are closed so my seven year old is at home with me on the days my wife is working. Thankfully this time round school was much better prepared and are setting work online and even have live lessons. My wife manages everything like a star for the multiple accounts and logins we've suddenly created for our daughter.

There was a moment last week where the maths work set for the class required logging into a service that they hadn't used since the last lock down. I texted my wife to check if she knew the password, but she was busy at work and said it's probably saved somewhere and she could find it when she came home. We got on with some other work, but then I realised that I could ask school - sadly reinforcing the stereotypical image of the dad who is devoid of any involvement with raising the children!

I don't know if it's just me but when helping my daughter through the work set for her I sometimes struggle to remember what is an appropriate level of detail for a seven year old. This week the maths work concentrated on money. One day there was a Power Point show with some extra "word problems". Our daughter struggles with these sometimes, but on the other hand if she has too may plain arithmetic questions she can start to get bored and a bit edgy. I told her that learning to be able to do the arithmetic is important because it is what you need to solve word problems - which are how maths comes up in real life, Paraphrasing something my father once told me.

Most of the problems were two step problems, i.e. An orange costs 23p, Alice buys 3 oranges and pays the greengrocer £1. How much change does she get back? In between helping her breakdown the question into the two parts (Step 1: Find the "cost of three oranges". Step 2. Find the change by doing £1 minus the "cost of three oranges") I would jump back to doing my work. Then suddenly, we got a really strange question, unlike the others.

There was an image of three coins: £1, 50p, 2p and the question was "Which of these coins is the odd one out? Explain your answer." I had a moment where I thought "That's strange, have they done Set Theory yet". We thought about it for a bit then started working on an answer. To help I told my daughter to try to say: "This coins is/has ..., but the other two coins aren't/don't". I avoided saying "These two coins are members of the set of ...., but the other one is not". I'd like to think that at some subconscious level this was because I wanted to think of the "odd one out" has having something unique and special about it, rather than "not being like the others" - she's not a teenager yet but the dramas with her friends are already starting!

Now for a seven year old she did pretty well. coming up with "The 2p is a copper coin, but the others are not" and "The £1 has two colours, but the others do not". We struggled a bit with the 50p. She came up with "The 50p is silver" but the £1 also has silver, so she tried "The 50p has corners" but on closer inspection so does the £1 - leading us to realise that actually the 2p is also unique in being the only circular coin. Suddenly we realised there was more than one answer to why a coin is the odd one out. I realised we could be really reductive and say "The 50p coin has a value of 50p, but the others do not" etc. Basically defining a set as "any coins that are not 50p" so the 50p is the odd one out. In fact in the answers provided by school they did say each of the coins could be considered the odd one out so long as it is justified correctly. The example they gave for the £1 coin was that "it has a value greater than 70p".

That's when I said to her, "What about this, the £1 and the 2p are odd ones out, the 50p is not".

Read this next paragraph carefully - I had to...

We made a set of things that are "odd ones out". The £1 is in the set for being the only coin with 2 colours, the 2p is in the set for being the only coin that is a circle. We can't find a reason to put the 50p in the set of "odd ones out". That means it is the only coin that is not in the set of "odd ones out"! Which means it is an "odd one out"! Aha! So the 50p coin can be in the set of "odd ones out" because it is the only coin that is not in the set of "odd ones out". But if the 50p coin is in the set of odd ones out then it's reason fro being and odd one out no longer exists - so it has to come out of the set - but if it is the only one out of the set then it can be in the set - but if it is in the set then it can't be in the set... but... but...

I recall about a year or so ago whilst we were reading Pinocchio at bedtime - the boy whose nose would grow whenever he told a lie - we came across a paradox. Whilst reading the story we spoke about the dangers of lying, but also how sometimes something may not really be a lie and one should be careful about their choice of words. What would happen if Pinocchio tasted some broccoli and really liked it, saying to you "broccoli is tasty" - you might try some and disagree. Did Pinocchio lie? Well not really, he may have phrased his statement incorrectly, offering an opinion as a fact. So perhaps he should have said "I find broccoli tasty". Whilst talking about this I asked her what would happen if Pinocchio said "My nose is about to grow". If his nose grows then he was telling the truth so his nose won't grow, if his nose doesn't grow then he lied, so his nose should grow. That time I think she just fell asleep. This time whilst I was getting excited about explaining the concept of a paradox in set theory she just stared at me blankly, then turned back to her work and did the next question.