We’re back with another challenge! This is in collaboration with Agent Venture who provide amazing online escape room experiences – make sure you check them out!

The Challenge

Agent Venture needs your top-notch bank robbery skills to help bring J. Bozo to justice for their numerous crimes.

The incriminating files against J. Bozo are locked behind a room with a keyboard and a 3×3 digital grid with letters on it… The passcode is in the form of a pattern which only a few employees know, can you work out what the pattern-passcode of the grid is?

The Solution

Let’s start by looking at Employee 1, we can see that there are multiple ways we can reach the typed passcode. See the images below to see some of the possible methods. How many ways of getting B, A, B, A, B can you find from Employee 1’s grid?

Similarly for Employee 2, we can see that there are also multiple ways in which we can reach the typed passcode. However, we can deduce that the pattern must start on the middle right square since there is no other starting point for C,C,B in a row (C,C,B,A,B).

Lastly, for Employee 3, there are also multiple ways of reaching the typed passcode C, A, C, B, C.

Using the information we gathered from Employee 2, we can deduce that if we start on the middle right square using Employee 1’s grid, then the next move is to go down to the bottom right square. If we instead try to go up to the top right square, then we reach a dead-end and end up with B, A, C even though the passcode for Employee 1 starts off as B, A, B (B,A,B,A,B).

It is still ambiguous what the next part of the pattern is by looking at the grids for Employee 1 and Employee 2. However, looking at the grid for Employee 3, we can see that the pattern must go through the middle square to get the last part of the pattern-passcode.

Putting all this information together and overlaying the pattern over the grid, we can deduce that the solution is F, I, H, E, D!

How did you do? Comment below how you reached the solution!

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?

Finally, you have your hands on the blueprints to the bank. You can see that on the blueprints there are only certain routes which you can take in order to avoid triggering the security alarms at 3:23AM (prime bank robbing time).

You need to plan your route carefully to get to the vault as quickly as possible and avoid detection. Luckily, someone has added the number of footsteps it takes along the routes on the blueprints!

First you need to go into the reception cabinet to obtain the keys to the HR office which contain employee records. Inside the records you will find the password to the computer of your accomplice who works in foreign exchange. Your accomplice has left you a key to access the cupboard by the Business Analyst’s desk. You need to find a file containing information about the top-secret floor – including the passcode to the vault.

Can you find a route that takes no more than 217 steps?

The Solution

Did you manage to find a solution under 217 steps? You’ve just solved a graph theory problem! Graph theory is the mathematical study of graphs which consist of vertices (or nodes, such as the Coffee Machine in the maze) which are connected by edges.

This problem in particular is known as the shortest path problem – we want to find a path between two nodes so that the total is minimised!

See the image below for the shortest path which takes exactly 217 steps.

The shortest route (with a total of 217) is: Entrance, Reception Desk (12), Reception Cabinet (9), Fish Tank (18), Microwave1 (7), Recruitment Materials (16), Employee Records (6), Training Materials (18), Head of HR Seat (10), Foreign Exchange (25), Microwave2 (36), Fridge (8), Frontend Engineers (11), Business Analysts (3), Data Analysts (19), Drinks Machine (14), VAULT (5).

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

All that planning has earnt you a break, so you decide to take a nap at the library. As you make your way there, you overhear some cops discussing when the last secret bank-heist occurred in the city.

Because you’re a nosey little thief, you manage to pickpocket 2 pieces of paper from them. You need to use the 2 notes to give you the information of the date of the last bank-heist…

When was the last bank-heist?

The Solution

The puzzle bit

The challenge is called The Library and upon closer inspection of the golden note, you may notice that the first letter of each word cycles through the letters ISBN: ISBN = International Standard Book Number!

The algebra bit

The 2nd note has a string of letters and numbers, but we do not know what the letters stand for. However we have some clues to solve them…

What is A?

The first equation (K + D)^{2} = A shows us that A must be a square number. The only possibilities between 1-9 are 1,4, or 9.

K and D must be also be between 1-9 and they must be different. The smallest possible choice here is that K and D are equal to 1 and 2 in some order, but A=1 or A=4 are too small to accept even the smallest valid values for K+D. Therefore A=9 which implies that K + D = 3 (incidentally also the smallest possible sum).

What are K and D?

Since we know that K+D=3, we have that either:

K=1 and D=2 or;

K=2 and D=1

If K=1, by the second equation C/F = 1, but this means that C=F which is a contradiction, since each letter must represent a different digit. Therefore, K=2 and D=1.

What is H?

Using the 3rd equation, with A=9 and K=2, we see that A-H=K implies that H=7.

What is C?

By the 4th equation: H+D=C, with H=7, D=1, we have that C=8

What is F?

By the 2nd equation: C/F = K, with C=8, K=2 we have that F=4.

Conclusion:

Now that we have found what each letter represents, we can see that the string of numbers is 9781444727296! This is the ISBN number for the book 11/22/63 (Stephen King) which is the date of the last bank heist! (Apologies for the Americanness of the date).

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?

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.

You arrive at the door to steal the bank blueprints, when you see Gary the guard.

“Do you have a drink?” he asks. “Guarding blueprints makes me so thirsty,”

He seems like a nice guy, but if you make him the perfect drink, perhaps you could put him to sleep for long enough to steal the blueprints.

You make your way to the bar and you see this:

The question is, which 3 bottles should you choose to make the perfect drink which will:

Put Gary to sleep

Won’t poison him

Be perfectly balanced

The Solution

There are many ways to do this, but here we can proceed through deduction.

We have 3 “elements” per bottle, which means that 9 elements will go into the draught in total. But we can only put 4 + 2 + 1 = 7 items in total. To account for the remaining 2 elements, we can deduce that they must be sunflowers.

Because we need 2 sunflowers, we must choose bottle F. The only alternative way to obtain 2 sunflowers is choosing both bottles B and E – but neither of these are purple, and since we can only choose a total of 3 bottles, even if the remaining bottle were purple, we would still be left with an extra sunflower; an unbalanced result.

To balance the 2 sunflowers, the remaining 2 bottles must be purple so we must choose from A, C, D, and H. Since the recipe only calls for 1 snail shell, we can deduce that we need bottle A as this is the only bottle to have only one snail in it.

Finally, to balance the recipe, we can conclude that the remaining 2 monkeys and elephant can only come from bottle H, so the final answer is that we should choose A, F and H.

The benefit of deduction is that it becomes clear that there exists no other solution.

Why is this algebra?

Because we have a system of 8 linear equations (the 8 bottles) with 4 variables (monkey, elephant, snail, sunflower; note that a purple bottle is equivalent to ” – sunflower”).

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