Happy Valentine’s Day everyone, I hope you are having a euphoric moment, but if not, you can try this algebra puzzle about maximizing the expression LUV + LU + UV + LV given L + U + V = 12.

What’s your favorite song about romance? Mine is A Thousand Years by Christina Perri.

I encourage you to play your go-to love playlist in the background, and give this problem a go. When you are ready, keep reading for the solution.

Solution

In love, and in mathematics, we often have to take a leap of faith and believe that things will work out. So let’s start with the following expansion.

Now we know L + U + V = 12, so we can rewrite the expression above.

We want to maximize LUV + LU + UV + LV, which happens to be part of the expansion above. So we can turn what we want to maximize into a product of 3 factors minus 13.

A consequence of the AM-GM inequality tells us that this is largest when all factors are equal, since L + U + V = 12, we can let L = U = V = 4, and that gives us

Did you get this right? And I wish you all a happy Valentine’s day full of love and grace.