The Natural Constant “e” and Finding The Best Girlfriend

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Recently, a close friend of mine has been troubled by whether or not to accept a girl’s confession of love. He sought my advice. In matters of love, I am a blank slate and cannot provide any valuable guidance. However, I suggest transforming this issue into a mathematical problem for consideration.

Suppose a person can have only one lifelong partner—although many people may not agree with this premise, let’s temporarily set it for mathematical modeling. How can this person maximize the probability of obtaining an outstanding partner in reality?

Once we establish this model, it becomes evident that this problem is similar to Socrates’ wheat problem. Around 2700 years ago, Socrates instructed his students to walk from one end of a wheat field to the other, without turning back, and to pick only one ear of wheat. The challenge was to select the largest ear of wheat without the option to switch or replace it.

To ensure the highest probability of picking the largest ear of wheat, we can approach this problem as follows:

1 2 3, 1 3 2, 2 1 3, 2 3 1, 3 1 2, 3 2 1

9.  Select the optimal probability as k/(k+1).

10. Similarly, if the i-th ear is the longest, the probability of selecting the best is k/i.

11.  Overall, the optimal probability of selection is:

12. The above equation can be approximated using the Newton-Leibniz formula.

13. Given that Z = k/n, the original formula is ln(Z)/Z. Since k < n, it’s evident that Z > 1. After deriving further, it was discovered that when Z = e, there exists a maximum value approximately equal to 1/e. 

Consequently, we obtain k ≈ n/e.

Applying this insight, we can address the friend’s dilemma regarding choosing a life partner. Suppose my friend intends to compare 30 girls during the process of finding his ultimate girlfriend. The optimal strategy for him would be to discard the initial 11 girls (approximately 30 * 37%) and establish a benchmark based on these 11 individuals. Starting from the 12th girl, he should wholeheartedly pursue anyone who surpasses the previously set standard. This approach represents the best possible solution for him.