January 2025

Prove or disprove:

\[2024^{2025} < 2025^{2024}\]

February 2025

An inverse sum triangle is a triangle constructed such that every $n$ sits above two integers, $a$ and $b$, whose absolute difference $|a - b|$ is minimal.

Some inverse sum triangles are not valid, as they cannot be completed. In the example below, we show one valid triangle, that of $10$. The invalid triangle, that of $18$, is invalid because the 2nd and 3rd entries (from the left) of the final row must both be 1, but $1 \not = 1+1$.

A failure rule is a pattern that guarantees an inverse sum triangle is invalid.

Please provide all possible failure rules, with proof of their universality (That they are true for all triangles) and their individuality (That they do not rely on another failure rule).