The values of $a$, $b$, $c$ and $d$ are 1, 2, 3 and 4, but not necessarily in that order. What is the largest possible value of the sum of the four products $ab$, $bc$, $cd$ and $da$?
I notice that the sum of the four products can be rewritten as $(a+c)(b+d)$ by expanding and collecting like terms. So I want to maximize the product of two sums of two numbers from the set $\{1,2,3,4\}$. I know that to maximize a product, I should make the factors as close as possible to each other. So I should pair up the smallest and largest numbers, and the middle numbers. That means $a+c = 1+4 = 5$ and $b+d = 2+3 = 5$. Then the sum of the four products is $(a+c)(b+d) = 5 \times 5 = 25$. 