Culture

The Stealthy Mathematics of Hearthstone

Blizzard's free-to-play collectible card game is easy to learn but difficult to master, and every play involves calculations you may not even know you're doing.
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Courtesy of Blizzard

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The situation in front of me looks dire. Four minions sit on the board, none of which are my own. They have a range of attack points, an onslaught that will surely deplete my remaining health points. A rope flies across the middle of the screen, igniting on the left and burning toward the right, signaling the impending end of my turn. I need to move fast. Suddenly the answer becomes clear: The health points of those minions are factors of two. I fling Defile out of my hand, dealing one damage to all minions, and putting the lowest health within elimination range. There’s just enough time to play Lord Godfrey, who pops off two damage to all others, repeating this over and over until all minions on the board are destroyed. I live to see another turn.

“Well played,” acknowledges my friend, also my current opponent, and my pride swells.

The game we’re playing is Hearthstone, a digital card game based on the long-popular MMORPG World of Warcraft. We’re currently in the middle of a 1:1 battle, pitting our decks against each other. It’s a time-consuming habit that’s admittedly a lot of fun. I’ve been enjoying Hearthstone since it first launched in March 2014, languishing in the escape it provides during my busy days. With new expansions and collectibles released each year, the game remains fresh and entertaining. But something else has come to my attention lately, and I think I have this game completely to blame.

I’m not merely a seasoned player, I’m a stronger mathematician.

As someone who occasionally struggles with math, this is huge. I’d sooner pull up a calculator on my phone than wrestle with a problem. Now, however, I find I’m capable of doing all sorts of calculations in my head. Maths that felt previously out of reach. A lot of solving for x. And when I went searching for a possible reason for these new skills, all signs pointed to the candy-colored card game on my screen.

Hearthstone is full of stealthy math, and players might not even realize it.

Gameplay is simple enough: Each player has an allotted number of mana (resources) to play a selection of cards in their hand. However, in the span of a typical 75-second turn, players calculate multiple combinations of the following: addition and subtraction, multiplication and division, value statistics, percentages, probability, and logic problems. That’s a lot of math in a short amount of time, yet the game pulls it off without ever feeling like you’re doing work. Sort of like executing a fancy skateboarding trick—you’re not actively thinking about the complicated mechanics involved, but they’re still there.

“For us, it’s about trying to find fantasies that match the card mechanics,” says Hearthstone lead game designer Alec Dawson. Animations and sounds cover up the tedium of the math. For example, the card Rolling Fireball deals eight damage to a single minion. However, if the minion only takes two damage before perishing, the excess rolls to the minions on either the right or left. The animation is exactly how you would imagine it in your own mind. “Seeing a big rolling fireball on screen makes it easier to understand how the damage is carrying over from one minion to the next,” explains Dawson. It’s visuals like these that so cleverly disguise the work you just did in your head.

Technically, my earlier board-clearing moves were merely basic subtraction problems. Written out, the equation to pull that off would look like H (health) - 1 - 2 x 4 = 0 (board clear). Let’s move on to more complicated stuff.

The Battlegrounds mode has a completely different set of rules, most similar to that of auto chess. Each lobby starts with eight players who eliminate each other until there is one winner. There’s a lot of random luck in this mode, but there’s a ton of probability involved too. In order to strategize a win condition, players need to already be thinking about turn five, six, and seven. And if your plan doesn’t pan out, you need to be able to pivot quickly. Add to all of this the unique special powers of your chosen character, strategic positioning, and multileveled tiers of minions, and players are cranking out sunk cost fallacy with every turn.

For example, a strong play here is to roll a powerful minion called Nomi, Kitchen Nightmare early on. However, the chance of getting a Nomi in any turn is tricky. Written out, the equation is 1-(1-Probability of Nomi)^Number of minions in a roll. Or 1-(1-9/1093)^5= 0.04. There’s approximately a four percent chance of getting Nomi on your turn (not factoring in the even slimmer probability of rolling multiple Nomi minions in one turn), but once you do, the benefits pile up with rewarding animations.

As you can see, it’s easy to play but complicated to master. Some of the pro esports players out there even have math PhDs. But do you need to be good at math to play the game? Definitely not.

Slysssa is a professional Hearthstone player and streamer for Team Liquid. Her specialty is the Battlegrounds mode. She’s also a former math teacher. “Math skills definitely help in different ways,” she notes. “Those who are good at arithmetic can count damage faster. Those who are strong in visual mathematics can also do well because they may have a better grasp for the strength of their board, the opponent’s board, and thinking ahead to how the board may look two-three turns down the road.” This isn’t math problems written out on a piece of paper, this is entirely visual. And unlike other tabletop-only card games, this is also completely virtual.

“Something we are quite cognizant of is the amount of tracking required as a player,” Dawson says, “So when we introduce cards with upfront complexity, we make sure there’s simplicity in the back-half or surrounding elements.” He’s referring to a mechanic in the game that involves shuffling additional cards into either your own deck or your opponent’s deck, with certain cards doing specific things depending on how many additional cards have been shuffled into that deck. Rather than having to keep track of this yourself, the card text will keep track for you.

Other times players will need to keep track on their own. Shuffling cards that deal damage when drawn into your opponent’s deck is another test of probability. If you’ve shuffled four cards in, and there are only eight cards remaining, it’s a 50 percent chance that one of your additional cards will be drawn on their next turn. The animations reinforce “shuffling,” and those visuals help to keep a mental count.

If all of this is sounding a bit overwhelming, fear not. Players are doing all of these cool calculations all the time, with whimsy and delight. Nothing ever feels like combat arithmetic, just an enjoyable game.

“We play-test all the time and gather feedback from our team at large,” Dawson notes. “Usually if something comes across as too complicated, it’s more about if the card was presented correctly or if the card text was clear enough, rather than the player failing to understand the concept.” A guaranteed way to keep the game fun. When there is that one card that feels more complicated to master than the rest, it can be even more rewarding to rise to the challenge.

Is Hearthstone out to teach its players lots of math? Not at all. But it morphed me from someone who broke out in a cold sweat whenever math was mentioned into someone who feels confident enough to want to teach it to everyone she knows. That’s a special kind of magic.

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