Monthly Archives: April 2025

Let $f(n)$ be the number of binary strings with length as $n$ that they do not contain a sub-string as “$010$” nor “$101$”. Prove that for $n\ge 3$: $f(n)=f(n-1)+f(n-2)$ 🔑 Proof: For $n=1$, we have $2$ possible binary strings, as … Continue reading →

Find the number of ways to write $24$ as the sum of at least three positive integer multiples of $3$. For example, count $3+18+3$, $18+3+3$, and $3+6+3+9+3$, but not $18+6$ or $24$. (PurpleComet 2023, Middle School, Problem 8) Click here … Continue reading →