Crack the Code: Unlocking the Secret Sauce Riddle

In the high-stakes world of culinary espionage, where restaurant rivalries run deep, a coveted secret lies hidden. Imagine yourself as a top-tier spy, tasked with infiltrating a web of deceit and misdirection to secure the ultimate prize: a competitor's secret sauce recipe. This isn't just about stealing a recipe; it's about outsmarting the competition and proving your strategic prowess.

The Mission: A Culinary Conundrum

The scene unfolds with a kidnapped chef, held captive by rivals seeking to extract the location of the famed secret sauce. Unbeknownst to them, you, the master spy from a competing establishment, are poised to capitalize on their scheme. Your mission is clear: decipher the clues, navigate the lies, and retrieve the recipe before anyone else.

Deciphering the Interrogation

From your concealed vantage point, you observe the interrogation. The captor believes the recipe is secured within a bank vault, specifically on the 13th floor, in a safe deposit box numbered between 13 and 1300. The interrogation proceeds with a series of questions, each designed to narrow down the possibilities:

• "Is the number less than 500?"
• "Is it a perfect square?"
• "Is it a perfect cube?"

Your challenge is that you can't hear the chef's answers, but you can discern when they are being truthful or deceptive. The interrogator, unfortunately, is easily misled, accepting the lies as truth. The clock is ticking as the interrogator believes they have the correct box number and dispatches agents to retrieve the recipe. You know they're wrong, and it's up to you to uncover the truth.

The Art of Deduction: Working Backwards

The key to solving this riddle lies in reverse engineering the information. The final question, regarding whether the number's second digit is a one, is the linchpin. By the time this question is asked, the interrogator has narrowed the possibilities to just two numbers: one with a one as the second digit and one without.

Your strategy is to identify answers to the preceding questions that result in only two possible numbers. The most restrictive clue is the one about the number being a perfect cube. This significantly reduces the options between 13 and 1300 to just eight possibilities.

Let's assume the chef truthfully confirmed that the number is a perfect cube. Now, consider the second question: Is the number a perfect square? If the chef had answered yes, the options would be limited to 64 and 729, the only numbers in the range that are both a square and a cube. However, neither of these has a 1 as the second digit. Therefore, the chef must have answered no to this question, allowing us to eliminate these two squares from consideration.

This leaves six numbers. Now, consider the first question: Is the number less than 500? A yes answer would leave four options, which is too many. A no answer, however, leaves two numbers greater than 500, one of which has a 1 as its second digit. The interrogator is under the false impression that one of these numbers is correct, but you know better.

Unveiling the Truth: The Final Solution

It's time to reconstruct the truth from the chef's lies and truths:

1. The chef lied about the number being greater than 500, meaning it is actually less than 500.
2. The chef lied about the number not being a perfect square, meaning it is indeed a square.
3. The chef truthfully confirmed that it is also a cube.

The only number under 500 that satisfies both conditions—being a perfect square and a perfect cube—is 64.

With the correct box number in hand, you swiftly locate the secret recipe, disappearing into the shadows before anyone realizes their mistake. Another successful mission concludes, proving that in the world of corporate espionage, intellect and cunning are the ultimate weapons. Sometimes, that’s just how the sausage is made.