Imagine you and a stranger are paired together for a little game. Now there’s some money up for grabs and you’re both given 2 choices; Share or Snake.

If you both share you both win £15 each. If one of you shares and one of you snakes, the snake will win £50 leaving the person who chose share with nothing. If you both pick snake, you both leave with nothing.

Would you pick ‘snake’ in the hopes of taking a bigger prize for yourself, or would you pick ‘share’ to share a smaller prize!? What would you do? What should you do? And why should we even care? In this episode, which is also the season finale, Bia shares some introductory game theory with Zoey by discussing:

The social-media experiment they both conducted through Instagram: “Snake or Share” The Prisoner’s dilemma (which is the original problem) The Traveller’s dilemma.

Acting like a “snake” i.e. picking the “dominant strategy” may give you control leaving you less susceptible to exploitation, but is it always the most profitable strategy? And what about the long-term implications of this?

How do dating apps work? And what are your thoughts on them? In this episode, Zoey shares how collaborative filtering works in dating apps such as Tinder, but also in Amazon. Bia shares how Hinge uses the Gale-Shapley algorithm (whilst butchering the pronunciation) to find your most compatible match. They discuss thoughts people shared via Instagram.

Further details of the maths and algorithms are shared via Instagram!

Time stamps: 0:47 – Collaborative filtering 13:55 – Gale Shapley algorithm 19:04 – Are dating apps are good/bad thing? Thoughts of Instagram followers 28:48 – Who do dating apps favour/ hinder? Thoughts of Instagram followers & some personal stories

In light of International Women’s Day 2021, Zoey and Bia interview Nina Chhita, a medical writer based in Canada. Nina brings together art and science by illustrating trailblazers in science who happen to be women.

00:10 Introduction 02:09 Quick fire quiz 03:15 What does it mean to be a medical writer? 05:31 Did you always want to study biology when you were younger? 08:10 Who were your role models growing up? 11:39 Which blue plaque story led to @science.unhinged and @nina.draws.scientists? 14:59 What was it about Rosalind Franklin that drew you to her story? 6:15 How do you get inspiration for the scientists you illustrate now? 18:26 How easy is it to find misinformation about less well-known women? 20:30 Which scientist you’ve illustrated has been most fascinating to you? 23:42 Has anything surprised you on this journey of science communication? 26:26 What are ways we can feature women in science to be more mainstream? 32:40 What would you change about the current curriculum to encourage girls to take more STEM-based subjects? 33:40 How much more progress do we need in the future and how do you think we can get there? 35:34 What makes a good scientist in your opinion? 37:45 What is planned next for nina.draws.scientist?

In this episode, Zoey and Bia answer questions submitted by listeners. These include what inspired them to make the podcast, Instagram page and blog, why a common person should learn maths, and how hard it is to get good at it.

Timestamps: 00:36 What inspired you to make this page and podcast?
02:00 Who are we?
02:49 How did we meet?
03:37 What inspired you to take maths at university?
04:38 What advice would you give yourself if you could go back in time 10 years?
05:25 What advice would you give to young girls who want to pursue STEM (Science, Technology, Engineering and Mathematics)?
06:30 Why do you think a common person should learn maths?
09:39 How hard is it to get good at maths?
12:40 What is missing from elementary school maths syllabus that would make maths more fun?
14:13 Are mathematical models underrated or overrated?
16:37 Do you have any maths book recommendations?
18:10 Does it annoy you that people think you are men online?
20:18 From your experience, what have you learnt about communicating maths ideas online?

Everything was going to plan, when you realise your friend is missing.

You notice three robot guards, and you think she may be disguising herself as one of them.

All robot guards are programmed exactly the same:

Each day is a truth day (so they must tell the truth the whole day) or a lie day (they must lie the whole day).

They all follow the pattern of one truth day followed by two lie days (T L L T L L T L L…)

They are not independent: They all have truth/lie days on the same days as each other.

The three guards say the following phrases:

Guard A: “In 2 days time, I will tell the truth”

Guard B: “Tomorrow, I will tell the truth”

Guard C: “One year* ago, I lied”

Which guard is secretly your friend?

*Assume one year = 365 days

The Solution

We know all guards will work in the same way, so we need to look for an “odd one out”. Let’s start by figuring out what day it is today. Is it a truth day, the first lie day (L1), or the second lie day (L2)?

Guard A

Guard A will tell the truth in 2 days time. Let’s look at the 3 cases of today being a Truth day, the first Lie day (L1), or the second Lie day (L2):

Truth day today: With the cycle T L L T L L…, in 2 days time it will be a lie day. But the guard said they would tell the truth, so this is a contradiction.

L1 today: With the cycle T L L T L L…, in 2 days time it will be a truth day. But the guard should be lying about it being a truth day, so this is another contradiction.

L2 today: With the cycle T L L T L L…, in 2 days time it will be a lie day. The guard said they will tell the truth which is a lie as it should be.

So Guard A must be speaking on the 2nd lie day: L2.

Guard B

Guard B will tell the truth tomorrow. Let’s look at the cases again, similar to above:

Truth day today: With the cycle T L L T…, we can see it is a lie day tomorrow. But the guard said they would tell the truth, so we have a contradiction.

L1 today: With the cycle T L L T…, we can see it is a lie day tomorrow. Since the guard is lying about telling the truth, this is consistent!

L2 today: With the cycle TL L T… ,we can see it is a truth day tomorrow: But the guard said they would tell the truth which is true on a lie day! A contradiction!

So Guard B must be speaking on the 1st lie day: L1.

Guard C

Either Guard A or Guard B must be the friend since we have shown that the day they are programmed to is different. This means there is no shortcut: We must work out which day it is today according to Guard C.

Guard C lied a year ago. This is clearly the hardest part of the puzzle. Let’s break this down into smaller chunks. We have a cycle of 3 repeating: T L L, T L L, T L L, etc.

If we go back 3 days ago, we land in the same place we are today. In fact any “multiple of 3” days ago, we get back to the same day.

366 is a multiple of 3, so if we go back 366 days, we land on the same day. We want to go back only 365 days, so if we add one day “forwards” from 366, we are actually saying that 365 days ago is equivalent to tomorrow!

Formally, we can write this as -365(mod 3) = 1 .

So one year ago is equivalent to one day in the future from our starting point (tomorrow). You could stop here since you can actually tell that B and C cannot co-exist given their mutually exclusive statements, but let’s conduct a final check that A and C are the guards. Recall that Guard A is speaking on the 2nd Lie day.

L2 today: With the cycle T L L T L L…, one year ago (tomorrow) was a truth day. The guard said they lied which is a lie as it should be.

Conclusion

To conclude, A and C are the guards, today is the second lie day (L2), and your friend is B! Were you able to deduce who your friend was?

Common errors

There were a couple of different elements to this puzzle making it challenging. As a result, very few people (out of ~60 submissions in total) sent in a correct solution on the first attempt. Here are some of the most common errors that were made:

Treating both lie days as the same. In reality, they operate differently due to their relative position in the cycle. Referring to “tomorrow” from L1 will give a different outcome than L2.

Not using a repeating cycle of 3 (T L L).

Calculating 365(mod 3) instead of -365(mod 3).

Calculating 365(mod 7), or using the days of the week in some way. This would lead to the wrong conclusion as we should think in terms of the 3-cycle.

Zoey and Bia discuss what zombie statistics are, why it’s hard for zombie statistics and facts to die and whether it is right for a wrong statistic to be cited even if it produces positive effects.

Introduction 00:15 – Lies, damned lies and statistics 01:48 – Zombie statistics definition Quiz and answers discussion 03:52 – Zombie stats or facts quiz 05:45 – Zombie stat/fact #1 – One in four people will suffer from mental illness/ depression in their lifetime 09:18 – Zombie stat/fact #2 – You need to drink eight glasses of water a day 12:18 – Zombie stat/fact #3 – People use only 10% of their brains 14:22 – Zombie stat/fact #4 – You need to walk 10,000 steps a day to stay healthy and fit 19:05 – Zombie stat/fact # 5 – The ban of plastic straws will massively reduce plastic waste in our oceans General discussion 26:26 – Discussion on why it’s hard for zombie stats/facts to “die” – beneficial information to people/companies and confirmation bias?
30:37 – Is it okay for a statistic to be wrong even if it has a positive effect?
33:45 – Making sure you understand the entire story of the statistic and taking it with a pinch of salt
35:29 – Conclusions

Bia and Zoey discuss some of the key mathematical concepts in voting, focusing on political elections in some Western countries, as well as Brexit.

Introduction 0:15 – Introduction on the voting system in the UK, with an example
4:27 – Condorcet’s paradox 6:00 – The French system
6:44 – The Australian system – Preferential/Alternative voting
7:42 – What defines a good voting system?
9:43 – How do we balance a good voting system with one which everyone understands
Arrow’s Impossibility Theorem & Instagram poll 11:40 – Arrow’s Impossibility Theorem
13:46 – Independent voting systems where Arrow’s theorem doesn’t apply.
15:00 – Instagram poll discussion: Tactical voting vs Voting for who you want
17:31 – Protest voting vs voting for who you want
20:20 – When it would be worth strategically voting “mathematically”
Brexit21:22 – Zoey exposing Bia as a “Remoaner”
22:50 – How Bia think the referendum should have been done.
General discussion 24:05 – Have you ever not voted?
26:12 – Should 16 year olds be allowed to vote? 2
27:29 – Accessibility in a voting system
The future of voting 27:54 – The future of voting
28:52 – The issues with a voting system which takes too long (NP-hard/ NP-complete)
30:00 – Dodgson’s voting method (Lewis Carroll = Charles Dodgson)
32:52 – Final thoughts

80% of voters are strategic: “Counting Votes Right: Strategic Voters versus Strategic Parties, Filippo Mezzanotti and Giovanni Reggiani” http://economics.mit.edu/files/11177