One thing you may have noticed about the trigonometric functions sine and cosine is that they seem to have no agreed upon definition. Or rather, different authors choose different definitions as the starting point, mainly based on convenience. This isn’t problematic or even particularly unusual in mathematics - as long as we can derive any of the other forms from any starting point, it makes little theoretical difference which we start from since they’re all equivalent anyway.

The Kaprekar routine is a simple arithmetic procedure which, when applied to four digit numbers, rapidly converges to the fixed point 6174, known as the Kaprekar constant. Unlike other famous iterative procedures such as the Collatz function, the somewhat arbitrary nature of the Kaprekar routine doesn’t hint at fundamental mathematical discoveries yet to be made; rather, its charm lies in its intuitive definition (requiring no more than elementary mathematics,) its oddly off-center fixed point of 6174, and its surprisingly rapid convergence (which requires only five iterations on average and never more than seven.

In 2020, the Zodiac 340 cipher was finally cracked after more than 50 years of trying by amateur code breakers. While the effort to crack it was extremely impressive, the cipher itself was ultimately disappointing. A homophonic substitution cipher with a minor gimmick of writing diagonally, the main factor that prevented it from being solved much earlier was the several errors the Zodiac killer made when encoding it. Substitution ciphers, which operate at the level of a single character, are children’s toys, the kind of thing you might get a decoder ring for from the back of a magazine.

Introduction Visualizing the results of a binary classifier is already a challenge, but having more than two classes aggravates the matter considerably. Let’s say we have $k$ classes. Then for each observation, there is one correct prediction and $k-1$ possible incorrect prediction. Instead of a $2 \times 2$ confusion matrix, we have a $k^2$ possibilities. Instead of having two kinds of error, false positives and false negatives, we have $k(k-1)$ kinds of errors.