"But... Will Optimus Prime like it?"

—Grapple to Hoist, The Master Builder

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First write-up: Oct 1, 2019

The newly revealed 🄺🄱 cards from Siege II finally give us the possibility to describe the ideal deck composition that maximizes our chances of activating Grapple's bot ability. This sounds counterintuitive at first, especially because of the peculiar color combination we're considering. In this set of notes we'll motivate this claim, and show an actual −even if non-competitive− deck composition which is both enabled by the new cards, and remarkably close to the ideal scenario.

In discussing Blowpipe, we've seen that the most efficient way of activating Grapple's ability is to play only two kinds of double-icon cards with non-overlapping colors. One color needs to be white, and we have access to a large amount of 🅆🄶 cards. Therefore, they will be the first double-icon card type chosen.

The second kind of cards can feature 🄾🄱, 🄺🄾, or 🄺🄱 color combinations. 🄾🄱 cards have been accompanied by a star cost since wave 2. But we've received both 🄺🄾 and 🄺🄱 star-free cards in Siege I and II. At the time of this writing, there are 4 of each kind, for a total of 12 copies available in either color combination. It's not a bad starting point.

🄺🄱 cards are preferred here because they have a better synergy with Grapple. Therefore, we'll consider a deck with only 🅆🄶 and 🄺🄱 cards, and evaluate the probability of activating Grapple's ability as a function of their relative ratio. But, before we do so, let's quickly examine the four 🄺🄱 cards we are going to include:

Steady Shot is the weakest of the four. But in a deck with no orange cards, we better take any static bonus compatible with our deck composition.

We're about to see that, with Tough 1 and 12 🄺🄱 cards, our rate of success in activating Grapple's ability is as high as 83%. With Smoke Cloak we can consistently count on a defense bonus of 5 or more.

Grapple's ability lasts until end of turn, and its effect is cumulative with its previous activations. With Point Position we can divert all attacks we're receiving on the wheel turn to Grapple, for no real (non-piercing) damage after the first attack.

Unflinching Courage is the ideal addition to the deck. We're about to see that Bold 2 can reduce our chances of failure in activating Grapple's ability to less than a negligible 10%.

With a deck including only 🅆🄶 and 🄺🄱 cards, we fail to activate Grapple's ability only when we flip all 🄺🄱 cards or all 🅆🄶 cards. Therefore, our rate of success is extremely simple to evaluate, and a very accurate approximation can be expressed in even simpler terms:

Probability (flipping battle icons of exactly 4 different colors)

≈ 1 − PWG4+Bold/Tough − PKB2+Bold/Tough .

The exact formula is not much more complicated, but it would require some additional notation. Therefore, it's omitted. PWG and PKB are our probabilities of flipping a 🅆🄶 or a 🄺🄱 card, respectively, and they can be both expressed in terms of the number nWG of 🅆🄶 cards in our deck. Therefore, our rate of success can be shown in Figure 1 as a function of nWG only. This plot shows the probability with no Bold/Tough, as well as with Bold/Tough equal to 1 (Smoke Cloak) or 2 (Unflinching Courage). Dashed lines show the approximated probabilities given by the previous formula, and how accurate that approximation actually is.

The main result shown in Figure 1 is that, with 12 🄺🄱 cards, we're already very close to the maximum probability we could possibly reach:

Figure 1

As a final word of caution, this build would require every 🅆🄶 card currently available. Therefore, these notes are not intended as a recipe for an actual deck. Our aim was to show the ideal deck composition aimed at maximizing the likelihood of activating Grapple's ability. The key feature we've outlined here is the requirement of diluting the color pattern we need to match over the smallest number of cards we need to flip, in this case 2. This way, we've quantified the highest success rate we can achieve, and a guideline to determine how effectively we're activating Grapple with an actual deck. With time, and an increased number of 🅆🄶 cards, we might reconsider this deck from a qualitative perspective, and try to approach these numbers via actual card selection.