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).

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”).