# Recreational Mathematics: Why Focus on Projects Over Puzzles

by Justin Skycak on

There's only so much you can hone your math skills by working on a problem that someone else has intentionally set up to be well-posed and elegantly solvable if you think about it the right way.

Cross-posted from here.

Personally, my most enjoyable and productive mathematical experiences while growing came from toy research projects. Some examples:

• When I learned about partial fractions decomposition in high school, I wondered if I could come up with a formula for the coefficients $c_{jk}$ in the partial fractions expression of

\begin{align*} \dfrac{b_0 + b_1 x_1 + b_2 x_2 + \cdots + b_n x^n}{(x-a_1)^{k_1} (x-a_2)^{k_2} \cdots (x-a_m)^{k_m}} = \sum_{j=1}^m \sum_{k=1}^{k_j} \dfrac{c_{jk}}{(x-a_j)^k} \end{align*}

by writing down and solving the system of equations in the general case, and also conducting numerical experiments to look for relevant patterns that I could prove and leverage in helpful ways. This turned out to be a very difficult and messy project but I did obtain one reasonably neatly-packaged result,

\begin{align*} \dfrac{x^n}{(x-a)^k} = \sum_{i=0}^{n-k} \binom{n-1-i}{k-1} a^{n-k-i} x^i + \sum_{i=\max(k-n,1)}^k \dfrac{\binom{n}{k-i} a^{n-k+i}}{(x-a)^i}, \end{align*}

and proving it gave me my first experience with double-induction.
• In my first year of college I got interested in neuroscience and I wondered if it was possible to, given a network of neurons, work out how the network's connectivity will change if you pick one neuron and repeatedly "activate" it with a pulse that ripples through the network. (Whenever two neurons activate, the connection strength between them changes according to known biological learning rules -- the simplest and earliest-discovered rule is "what fires together wires together," known as Hebbian learning, but since then more nuanced rules have been discovered like spike-timing-dependent plasticity, which takes temporal directionality into account.)

Similar to the previous problem, my approach was again a combination of working out math by hand and also running computer simulations to look for helpful patterns. The problem was again very difficult and messy, but again I really leveled up my skills by grappling with it and I discovered plenty of interesting things along the way.