Some may argue that designing a board game is not art, it is simply the process of dressing up a mathematical structure with a theme and the few tweaks that make the game interesting can be easily added during play-testing. If you play Uwe Rosenberg's Agricola you will most likely have the peculiar feeling that the game has been engineered o be perfectly balanced, still offering enough options that no two games are alike.
The whole genre of German (or Euro) games are usually almost luck independent, they challenge only the wit and skills of the players and the more you play the better you are. In a typical Euro-game, it is quite unlikely for an experienced player to be defeated by a newbie. Therefore, we may safely say that most Euro-game have an engineered engine, a mathematical model which is usually nicely hidden behind a village or town building like theme to make it more attractive to light gamers and families.
However, those of you who have played lots of Euro-games know already that there are hardly two games alike - I am talking here about the good games, not about cheap clones - and that every game challenges you in a different way. If you know how to win Agricola, you won't be able to apply the same 'algorithm' for Puerto Rico, Trajan or Ora et Labora. As a game designer myself, I must argue against the assumption that most Euros are a nice cover for simple (or complex) maths. I am not saying that there isn't a core based on an algorithm, most good games have that, I am simply stating that there's a lot more to making a game than the maths behind it.
You may have noticed that the best games out there are innovative in at least one way. This is where art or inspiration comes in. It's not enough to be a good engineer and apply optimization algorithms to have a good game. You might end up with a perfectly balanced game, but to achieve the holy grail (new + fun + balanced) you need a drop of ... something else. Innovation is not as easy to describe and quantify, but I will give it a try. When Puerto Rico was launched, it brought something that most player have not seen before, the mechanism of the common action chosen by the active player, executed by everyone. This was the new and brilliant touch that pushed Puerto Rico to the top of gaming charts. In Terra Mystica (best Euro I have ever played) it's difficult to pin-point a single innovation that makes the game great. If I had to choose, I would go with the way the theme integrates with the game mechanisms. I don't know how the authors worked this game, but I notice the result. Terra Mystica does not feel at all like two separate parts, the algorithm and the theme, it feels like an inseparable integrated body. Each race has one or more unique abilities and it also comes with deviation from the standard costs vs. advantages. However, each race feels so solid that I could not have imagined it having different abilities. Designing this game is definitely a combination between art and engineering, because the game a whole is balanced without being boring and each race is unique without being overpowered.
Let's take a look to a different genre, the Ameritrash games. In my opinion, most of these games are leaning more towards the art side. To make a successful game of this genre, you need inspiration above all. You need to have a story, an universe that sucks people. Of course you will also need solid game mechanisms, but the theme dictates the mechanisms and not the other way around.
Regardless of its type, whether it is thematic, family or strategy, any game is a mix of inspiration and hard work. The hard work is usually on the engineering side, when you bust you brains to balance resources or to quantify different paths to victory but the small tweaks that make a good game great are always the fruit of inspiration. In my opinion, making a game is just as much art as it is engineering because you need to be creative to make a board game but you also need to develop a technique of development to make the game playable and balanced.