One of the top chefs from Pasta Palace has been kidnapped by operatives from Burger Bazaar hoping to learn the location of their secret sauce recipe. Little do they know that a third party— Sausage Saloon— has sent you to take advantage of the situation. As their top spy, your skills range from infiltration and subterfuge, to safecracking and reading faces for signs of deception.
You’ve tracked the captors to where they’re holding the chef prisoner. From your hiding spot, you can see him on the other side of the glass, while in front of you an interrogator wearing headphones speaks into a microphone.
“We already know the recipe is on the 13th floor of the bank vault, in a safe deposit box numbered between 13 and 1300. Now tell us… Is the number less than 500?”
You can’t hear the chef’s answer, but you can see that he’s lying. The interrogator, however, falls for it. He follows up by asking, “Is it a perfect square?” Again you can’t hear the answer but can tell the chef is lying, while the interrogator takes him at his word. He then asks, “Is it a perfect cube?” This time the chef answers truthfully.
The interrogator thinks for a minute and says, “Good. Now if you just tell me whether or not the number’s second digit is a one, we’ll be done here.” But as the chef starts to answer, the interrogator stands up, blocking your view. Within moments he rushes out of the room, announcing that he’s got the answer and is sending agents to retrieve the recipe.
You know that the Burger Bazaar people have the wrong box number. But can you figure out the right one and retrieve the recipe yourself?
Pause the video to figure it out for yourself.
Answer in 3
Answer in 2
Answer in 1
The key here is to work backwards. We don’t know what the chef answers to the final question or whether he answers truthfully. But we do know that by the time the interrogator asks it, he’s narrowed the options down to two numbers– one where the second digit is 1, and one where it isn’t.
Our goal, then, is to find answers to the previous questions that lead to just two possibilities. Of the three constraints offered, the one that narrows our options the most is if the number is a perfect cube. That leaves us with only eight answers between 13 and 1300. So let’s assume the answer to the third question was a truthful YES.
Now, let’s look at the second question. If the chef answered YES to the number being a perfect square, it would narrow the interrogator’s options to just 64 and 729– the only numbers in our range that are both a square and a cube. But neither of these has a 1 as the second digit. So the given answer to the second question must’ve been NO. And that also means we can eliminate these two squares from the interrogator's list, leaving only six numbers.
Now for the first question, which allows us to divide this list. If the chef answered YES to the number being less than 500, we’d have four options, which is too many. But a NO leaves us with two numbers greater than 500, one of which does have a 1 as its second digit.
We don’t know which of these numbers the interrogator thinks is correct. But that doesn’t matter– remember, his conclusion was based on lies. You, on the other hand, are now in a position to reconstruct the truth.
First, the chef said the number was greater than 500 but lied, meaning it’s actually less than 500. Second, the chef said it wasn’t a perfect square but lied again, meaning the number is indeed a square. And finally, he truthfully confirmed that it was also a cube. And as we’ve already seen, the only number under 500 that’s both a square and a cube is 64.
You find the secret recipe and are gone before anyone’s the wiser. Corporate espionage is not an easy game— but sometimes, that’s just how the sausage is made.
