Tag: algebra

Challenge #11 – The Library

Challenges

The Challenge

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).

Challenge #9 – Poison

Challenges

The Challenge

Difficulty: 1/5

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”).