Problem 1: The Restaurant Bill

Consider this scenario:

At a restaurant, the total bill amounts to x dollars and y cents, where both x and y are integers greater than 1, satisfying the equation ( y^{4/3} = x^{5/6} ). What is the total bill?

I requested three data scientists to solve this. They all turned to code, calculating ( y^{4/3} ) for values of y from 2 to 99 and then checking if the result raised to ( 3/4 ) was an integer. This process, although effective, was time-consuming.

However, it can be resolved simply with paper and pencil. By converting the indices to a common denominator of 6, we conclude that ( (y^?)^{1/6} = (x^?)^{1/6} ), leading to ( y? = x? ). Since y must be below 100, we can use powers of 2, resulting in a bill of $256.32.

Problem 2: Finding n

Next, consider the equation:

Given that n and ( sqrt{n-2} + sqrt{n+2} ) are both integers, determine n.

Two data scientists opted to simulate values of n. They quickly found a solution, with one using a simple guess-and-check method. However, this approach only partially addresses the problem, as it doesn't prove that n has a unique solution.

To confirm the solution and demonstrate its uniqueness, we note that ( n-2 ) must be positive, implying ( n geq 2 ). We also find that ( (sqrt{n-2} + sqrt{n+2})^2 ) must be an integer. By expanding, we derive that ( 2(n + sqrt{n^2-4}) ) must also be an integer. It follows that ( sqrt{n^2-4} ) must be an integer as well. Consequently, for n less than or equal to 2, we conclude that n must equal 2.

Problem 3: The Salamander Conundrum

Here’s the final, and arguably the most challenging, problem:

On a deserted island, 30 salamanders exist—15 red, 7 blue, and 8 green. When two salamanders of different colors meet, they both transform into the third color. When two of the same color meet, they each change into one of the other two colors. Is it possible for all salamanders to eventually turn red?

Once again, the data scientists turned to simulations, with some running random trials up to a million times without success. Another attempted to program a loop that would terminate when all salamanders turned red, but this process seemed endless.

This illustrates the distinction between statistics and pure mathematics. A statistician might conclude that it’s improbable for all salamanders to turn red based on their simulations, as they reduce probabilities to a point where the null hypothesis is deemed "unlikely." However, this does not constitute proof.

To establish proof, we can analyze the remainder of each color when divided by three: red is 0, blue is 1, and green is 2. Notably, when any two salamanders meet, the remainder structure remains unchanged, preserving the balance of 0, 1, and 2. Therefore, if all 30 salamanders were to turn red, the remainder would be 0, 0, 0, which is impossible under the established rules.

In conclusion, how would you tackle these problems? Do you believe that the rise of data science is leading to a decline in our mathematical thinking? I welcome your thoughts in the comments.

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