In-Depth Turn Analysis #8: Arithmetic

Introduction

About This Week’s Deck

This weeks deck is:

[spoiler]This week we are running ‘Freeze Mage’. The exact deck-list does not matter too much (due to the nature of the position we are going to study). So basically, we are playing a deck a bit like this one:

[/spoiler]

You can learn more about the deck by clicking on the following links:

[spoiler]Camzee’s Guide[/spoiler]

The Situation

Okay, so let’s look at the position shall we?

In actual games, positions have context beyond the deck you are using. In an effort to make this series more realistic and less ‘puzzle-like’ let’s talk about a few cards that have been used in the game so far and any other little bits of information that seem relevant.

YOUR MANA: 10

NUMBER OF CARDS IN HAND (OPPONENT): 6

YOUR HAND: {10 Cards} Pyroblast, Ice Block, Ice Lance, Fireball x5!, Frostbolt, Acolyte of Pain

KEY CARDS USED (YOU): Since we have entered into Fatigue, ‘key cards used’ is irrelevant

KEY CARDS USED (OPPONENT): I don’t know (but probably not relevant anyway).

ANY OTHER NOTES: Pyroblast costs 9 mana. The Secret in play is Ice Block. Acolyte of Pain was drawn at the start of this turn, therefore, we take 1 Fatigue damage on our next turn.

Armed with this information, you must now start to think about the needs/peculiarities of both the position and the match-up. Get busy thinking!

Smashthings’ Analysis

[spoiler title=”Smashthings”]This week my analysis is going to be a little on the shorter side, this is because once you figure out the problem there is clearly only one play that is even worth mentioning. So, this week I’m just going to tell you what the play is and then explain why this is the only possibility. But before I tell you, I would strongly recommend that you try and solve it yourself first. I’d make that recomendation partly because solving puzzles is fun, and also because trying to solve the position (whether successful or not) will make you a better player.

Okay so without further ado, what’s the play?

Fireball our own face! (Yes, you heard me right), and then Fireball + Frostbolt opponents face.

Okay, so why is this the best (and only) play? I think it is best to sub-divide the solution into two smaller sub-problems:

Why Frostbolt + Fireball the enemies face
Why Fireball our own face.

Let’s tackle each of these sub-problems in turn.

Problem (1)

At 16 life there is no combination of cards in our hand that can outright kill the Mage; the best we can do is 15 which is one-off lethal. Moreover, given the fact that we are in fatigue and that the enemy Mage has an exceptionally strong board (which we cannot remove) it is clear that our only chance of winning is to go face with everything. Also note that Fatigue damage ignores Ice Block, this means that we have this turn and the next to win the game.

Okay, so we are going face with all our damage spells, but why are we using Frostbolt this turn (as opposed to saving it for Ice Lance)?

The Answer: The most damage we can do over two turns is 28: Fireball x4 + Frostbolt + Hero Power. The other combo (Frostbolt + Ice Lance + Fireball x3 + Hero Power x2) is one less at 27. Thus, while it may initially seem tempting to hold onto the Frost Bolt in order to deal damage with Ice Lance the reality of the situation is that such a play would be incorrect because you are never going to be able to find the time to play Ice Lance in this game. With this said, the chances that one damage will make the difference are rather slim, but it could (at least in theory) make the difference. And in my opinion, it is simply good practice to find the most optimal play.

Okay, so that is the boring/easy bit out of the way, now let’s study why you need to fireball your own face!

Problem (2)

Once you understand the problem seeing this play is not that difficult. So what is the problem? Well, the problem is that we lose the game if we drop to 1 health (regardless of whether Ice Block is triggered or not) due to fatigue damage.

We also obviously lose to a Kezan Mystic, and/or should the opponent counter our burn (e.g. with his/her own Ice Block) but these possibilities are things that we cannot do anything about and so therefore it is futile to worry about them. Since we can’t do anything about these possibilities we must simply forget about them and ‘play-around’ what we can.

It should be clear to everyone reading that our our objective for this turn is twofold:

Deal enough damage such that we have lethal on the following turn.
Make it difficult for our opponent to get us to 1 life.

Objective (1) is easy to accomplish, we can do that by flinging a spells at the enemies face, but objective (2) requires a lot more careful consideration. Moreover, due to objective (1), we have an maximum of eight mana to solve objective (2). Okay, let’s get started!

We start the turn at 9 life and so the question we need to ask ourselves is; “How easy is it for our opponent to put us to 1 life (i.e. deal exactly 8 damage)?”

As it turns out, the Mage has a wide range of ways to do damage; notice that the minions on board have a range of different attack values {3, 4, 5}, and then there is Hero Power (1) and Fireball (6). Thus the attack values the Mage has are: {1, 3, 4, 5, 6}. The puzzle to solve is how you can add those values to make 8. By my count, the Mage has three options: (a) 1+3+4, (b) 5+3. The third option, (c), is to attack with Azure Drake (4), and then cast a spell on one of his/her own minions to buff Mana Wyrm to 4 attack (4+4 = 8).

When I first looked at the position, I was trying to Fireball Face (to fufill objective 1) and then tried to freeze two minions with Frostbolt and Icelance in order to stall the game for one more turn (which is objective 2). Hence my initial solution was to Freeze Loatheb and Azure Drake. This removes options (a), (b), and (c). But it unfortunately loses to at least two other possibilities:

Frostbolt + Hero Power + Mana Wyrm.
Fireball Azure Drake, Fireball Loatheb, attack with Antique Healbot and the buffed Mana Wyrm.

And with an Archmage Antonidas on the board, play#2 above is extremely likely. In short, trying to Freeze enemy minions is not likely to work (assuming an opponent smart enough to spot play #2) and this is simply because of the wide array of attack values the Mage has to play with.

It is perhaps worth pointing out that if all the minions where 6 attack our enemy would have a much harder time winning the game since dealing exactly 8 damage in such a case would be difficult for them: Frostbolt/Fireball + minion attack is too much damage, Hero Power + 6 attack minion is too little damage, and so on. So yes, it really is the range of damage that the Mage has that is the problem with trying to freeze the board.

Okay, so we know that at 9 life our opponent can easily deal 8 damage to us, even if we Freeze minions. Thus, if we cannot prevent the damage we must ‘adjust’ our life total this turn. Since Healing is not currently possible, damage is the only ‘adjusting’ we can do. This then begs a very simple question: “What combination do we use?”

Why not Frostbolt?
Why not Frostbolt + Hero Power?
Why not Frostbolt + Icelance?
Why not Frostbolt + Icelance + Hero Power?
Why not Hero Power by itself?

This looks like we have lots of questions to answer. But this week I have decided to be a little bit lazy and make you work for it. 🙂

The problem (in generalised form) can be written in the following way:

If (Dmg >= (X-Fd) AND Dmg < X) Then the enemy wins.

^ Where ‘Dmg’ is the damage they have. ‘X’ is the Freeze Mage’s Lifetotal and ‘Fd’ stands for Fatigue Damage.

Now that I have generalised the problem, you should be able to go through the questions and convince yourself that ‘Fireball Face’ is the only option. With that said, I’ll quickly answer one of the questions posed, just to show you how: “Why not Frostbolt (Fbolt) + Hero Power (Hpow) our own face?”

Fd = 1, X = 9, Dmg = {1,3,4,5,6}

X – (Fbolt + Hpow) = 9-(3+1) = 5
(X- (Fbolt + Hpow)) -Fd = 5-1 = 4.
Therefore, If the enemy has a way to deal 4 dmg (without doing more than four damage) , we lose.
The enemy has Azure Drake. Therefore we lose.

Let’s do the same analysis for Fireball face:

9-6 = 3
3-1 = 2
Therefore, If the enemy has a way to deal 2 dmg (without doing more than Two damage), we lose.
The enemy (probably) cannot do this! Therefore we win?

Remember earlier I said that the set of Attack Values the Mage has are: {1,3,4,5,6}. Notice that the number two cannot be constructed given this set of numbers! And so therefore we ought to survive the turn (barring any surprise tech card from our opponent, such as an Elven Archer, or Ironbeak Owl on Mana Wyrm). And now that we have found a way to survive we actually stand a chance at winning the game (provided our opponent has no defence to our Fireball x2 + Hero Power follow-up play).

In my mind, if these In Depth Turn Analysis series did nothing more than simply explain the specifics of a given position I don’t think the series would be very helpful. Therefore, when explaining plays I go out of my way to highlight the general principles and the theory behind the position. If you understand the principles, you can apply the concepts more easily to your own games.

If you play Freeze Mage these sorts of positions are in fact surprisingly common. I hope that by taking the time here to explain the theory, the logic, and generalise the principles that you should be able work out what to do if you ever find yourself in such a position.

Smashthings signing out![/spoiler]

…In the End…

In this section we show/tell you about what actually happened during the game. Click on the spoiler to find out!

MINOR UPDATE: Judging from the comments, I need to make something clear: THE POSITION WAS NOT TAKEN FROM ONE OF MY OWN GAMES!!

[spoiler]

[/spoiler]

Conclusion

And that conclude’s this week’s instalment of In-depth Turn Analysis. Feel free to leave a comment about this weeks instalment.

And if you like it, don’t forget to leave a thumbs up!