The Angry Statistician

A tribute to Martin Gardner . For which sets of positive integers does the sum equal the product? For example, when does $x_1 + x_2 = x_1\cdot x_2$? It's easy to see that this is only true when $x_1 = x_2 = 2$. In the general case our equality is $\sum_{i=1}^{n} x_i= \prod_{i=1}^{n} x_i$. We can rearrange terms to give $$x_1+x_2+\sum_{i=3}^{n} x_i= x_1\cdot x_2\cdot \prod_{i=3}^{n} x_i,$$ and this in turn factors nicely to give us $$\left( x_1\cdot \prod_{i=3}^{n} x_i - 1\right)\cdot \left( x_2\cdot \prod_{i=3}^{n} x_i - 1\right) = \left( \prod_{i=3}^{n} x_i \right)\cdot \left(\sum_{i=3}^{n} x_i \right) + 1.$$ How does this help? Consider the case $n=5$, and without loss of generality assume $x_i \ge x_{i+1}$ for all $i$. If $x_3=x_4=x_5=1$ our factorized equation becomes $$(x_1-1)\cdot(x_2-1)=4,$$ with the obvious solutions $x_1=5, x_2=2; x_1=3, x_2=3$. The only remaining case to consider is $x_3=2$, as any other case forces  \( \sum_{i=1}^{n} x_...