Instead of directly simulating our flips, we will first try to work out the probabilities of the possible flips in an analytic way. This will give us a better understanding of the role played by the different kind of cards in our deck. It would have been possible to do the same with Metroplex, but much more cumbersome, because of the combinatorial increase of possibilities caused by Bold. We will eventually simulate our flips, but mainly to verify that our more conceptual understanding is correct. As a bi-product of this we will also quantify the error induced by some approximations that we might mistakenly consider more impactful than they actually are (i.e. difference between probabilities without and with replacements).

Let's start by reconsidering the problem of matching a color pattern in a more general way. We need to flip one of the right combination of colors. Each color, or color combination, has its own probability of being featured on the top card of our deck, and that depends on how many cards with that combination we play. If we need to put several cards together to match a pattern, then our success rate will be the product of their individual probabilities of being on top of our deck. The more cards we need to match a pattern, the more products we need to take, and the lower the resulting product will be (e.g. 50% of 50% is 25%, but 50% of 50% of 50% is 12.5%, etc.). This means that the best way to increase our chances of matching a pattern is by reducing the number of cards we need to put together. This is, for example, why double-icon cards are so important in Metroplex. They reduce the number of orange and blue cards we need by a factor two. In a case like Metroplex, this is evident. But there are less evident cases, like Grapple. If it wasn't for their effect, we would never play a lot of 🄾🄶 and 🄱🄶 cards in a Grapple deck when, ideally, we only need 🅆🄶 and 🄾🄱 cards to match our pattern. Of course, with Grapple, we face the problem posed by the scarcity of ★-free 🄾🄱 cards, but we can still achieve about 70% success rate (without Bold!) by investing into some ★-cards, and refusing to dilute the distribution of our colors over 🄾🄶 or 🄱🄶 cards. 

With Blowpipe we deal with the additional complication posed by the fifth battle icon, but the argument still holds, and we can achieve a similar result. We need as many 🅆🄶 cards as we can afford, and leave the remaining colors to the other cards. One reason for that is the abundance of 🅆🄶 cards. They can constitute half our deck if needed. This way we can allocate about half color pattern on a single color combination, notably the one that makes us flip more cards. The remaining half patter will be on about any of the other cards.

In the following, nGW will be the number of 🅆🄶 cards in our deck. The other colors, orange (O), blue (B), and black (K) do play a symmetric role. Therefore, we'll include equal number of them, and only through double-icon cards. We will call nXY the common number of 🄾🄱, 🄾🄺, and 🄱🄺 cards in our deck. Therefore, in a 40-card deck,  nWG +3 nXY = 40. This is where we enforce the symmetry that allows us to reduce the number of parameters we need to deal with.

Flips

Let's consider the possibilities for our flips in a deck like the one we are building. We're not including single icon cards. Therefore, we don't count on Bold, and consider only the case in which we attack by flipping two cards. We either flip at least a 🅆🄶 card with our initial flips or we don't. In the first case we end up flipping four cards, in the second case just two.

Two cards: Our two cards are not 🅆🄶, and they either have the same color combination or they don't. In the first case we flip two different colors, in the second case we flip three. We cannot flip four or five colors when we only flip two cards.

Four cards: We're flipping at least one 🅆🄶 card. Therefore, we cannot flip exactly three colors, and we flip just two colors only if all four cards are 🅆🄶. We flip four colors when the remaining three cards have the same two-color combination, when they're not 🅆🄶 themselves. Lastly, we flip five colors when we have at least two different non-🅆🄶 cards.

We can summarize the different possibilities as follows:

Probabilities

The probability of flipping a 🅆🄶 card from the top of our shuffled, 40-card deck is PWG = nWG/40, i.e. the relative number nWG of 🅆🄶 cards in our deck. Similarly, the probability for the top card to be 🄾🄱 is PXY = nYX/40, and it's the same for 🄾🄺 and 🄱🄺 cards, as we're building our deck with equal numbers of 🄾🄱, 🄾🄺, or a 🄱🄺 cards. These probabilities change when we flip more than just one card. For example, if the first card we flip is 🅆🄶, then our chances of flipping a second 🅆🄶 card becomes (nWG − 1)/39. In the same scenario, the probability that the second card is 🄾🄱 is nYX/39. It is possible to work out these probabilities, and the result is conceptually simpler than lofty names like hypergeometric distribution would suggest. But we will follow an even simpler approach in these note. We will pretend that additional flips happen after we reshuffle every card we've already flipped back into our deck. Therefore, our probability with replacements of flipping a 🅆🄶 card will always be PWG. The probability of flipping an 🄾🄱 card will always be PYX, etc. We will check how impactful this simplification is by comparing our result with an actual simulation of our flips, as simulations easily take into account the effect of being depleting our deck as we flip multiple cards. 

With this simplification in mind, we can now write simple formulae for the probabilities of the different flips listed in table 1 in terms of PWG and PXY :

P ( 2 colors with 2 cards ) = 3 PXY2

P ( 3 colors with 2 cards ) = 6 PXY2

P ( 2 colors with 4 cards ) = PWG4

P ( 4 colors with 4 cards ) = 12 PWG3 PXY + 15 PWG2 PXY2 + 6 PWG PXY3

P ( 5 colors with 4 cards ) = 30 PWG2 PXY2 + 48 PWG PXY3

These probabilities are derived by counting the individual ways of flipping a given number of colors (boldface numbers). They're boring, but not complicated to derive. Let's just go through a simple case together, and evaluate P ( 2 colors with 2 cards ). We've seen that we can flip two 2 colors with 2 cards only if the cards share the same, non-🅆🄶 color combination. As there are three different combinations of this type, there are three different ways of obtaining this result. The probability of drawing each one of the two copies is PXY. Therefore, P ( 2 colors with 2 cards ) = 3 PXY2 . The other probabilities might require more work, but are derived in a similar way.

Combining them together, we can write down the probabilities of flipping any number of colors:

P ( 2 colors ) = P ( 2 colors with 2 cards ) + P ( 2 colors with 4 cards )

P ( 3 colors ) = P ( 3 colors with 2 cards )

P ( 4 colors ) = P ( 4 colors with 4 cards )

P ( 5 colors ) = P ( 5 colors with 4 cards )

As PWG + 3 PXY = 1, PXY can be rewritten in terms PWG of as PXY = ( 1 − PWG ) / 3 , which allows us to express these probabilities in terms of PWG only (formulae omitted, just replace PXY = ( 1 − PWG ) / 3 wherever it appears), and plot them as functions of nWG. The result is shown in figure 1.

Figure 2 shows how well our analytical approximations compare to the more precise result obtained by averaging  over 100'000 simulations of our flips (dashed lines). Each individual panel in figure 3 refers to a fixed value of  nWG and, therefore, of nXY. The first conclusion that we can draw is that neglecting deck depletion is not a bad approximation, as long as we are flipping a limited number of cards from the top of our entire deck.

And now that we know that we can trust our analytical result, we can use it to better understand the anatomy of our flips.  

With reference to figure 2, let's consider the far left region of the plot, corresponding to a very small number of 🅆🄶 cards. In this case, we almost always flip only two cards. Therefore, we flip 2 or 3 colors most of the times. We more often flip 3 colors, as we need to flip two non-🅆🄶 cards with the same color combination to flip just 2 colors.

When the number of 🅆🄶 cards increases, so does our probability of flipping 4 cards, and, therefore, 4 or 5 colors. We never flip 3 colors with 4 cards. Therefore P ( 3 colors ) is a decreasing function of nWG. The same is not always true about P ( 2 colors ). But let's first consider P ( 5 colors ). Its maximum corresponds to nWG ≈ 16. This is also the turning point for P ( 2 colors ) because, as we increase nWG, we also increase our chances of flipping 4 🅆🄶 cards. 

We can still capitalize on the slower rate at which  P ( 2 colors ) is increasing, when compared to P ( 4 colors ). This being true at least until nWG  reaches approximately 28. That's when our chances of flipping 4 colors are at their peak. After that, our chances of flipping double-🅆🄶 are just two high. Eventually, P ( 2 colors ) becomes 100%, and any other probability drops to zero.

We can conclude that the optimal number of 🅆🄶 cards must be somewhere between 18 and 28. And a good way to identify it is trying to maximize our chances of flipping either 4 or 5 colors. As 

P ( 3 or less colors ) + P ( 4 or more colors ) = 1 ,

maximizing  P ( 4 or more colors ) also corresponds to minimizing  P ( 3 or less colors ). Clearly

P ( 3 or less colors ) = P ( 2 colors ) + P ( 3 colors ) 

and

P ( 4 or more colors ) = P ( 4 colors ) + P ( 5 colors ) .

Figures 3 and 4 are the analogous of figures 1 and 2 for these two sums, and they show the result in very simple terms: The optimal number of 🅆🄶 cards would be about 22-25. 70% of times it would correspond to an attack bonus of +5 (plus the the number of orange icons we flip).

P ( 4 or more colors ) is barely affected by the inclusion of Quartermaster, and worsened by an acceptable 6% by a mono-color, double-icon card.  The reason for the stability of our result is the fact that we are keeping the most important features intact: 1) Color pattern split between two sets of interchangeable cards; 2) About half our cards featuring a white icon. 

The two cards considered here are worth being included regardless of our team of characters. Any additional modification will instead depend on the rest of our team. Therefore, the next section will feature some good teammates for Blowpipe.

Teammates

Slipstream, Strategic Seeker. Slipstream is among the first characters coming to mind, because of how similar her bot ability is to the one printed on the compression cannon. Slipstream's ability adds +3 attack when we flip 3 or more colors. Nothing more if we flip more colors, nothing at all if we flip less. She's not really our best partner for Blowpipe, as we're about to see, but still worth considering.   Figure 7 shows the probability distribution of her attack total in bot mode, in a deck with 19 🅆🄶, 3 🅆 (Quartermaster), 3 🄾🄱, 6 🄾🄺, 6 🄱🄺, and 3 🄾🄾 cards (Peace Through Tyranny). All other charts in this section assume this same color composition, and their corresponding characters in their bot modes. Slipstream has a considerable chance of attacking for less than 10, including cases when we only flip white and green cards, or two 🄱🄺 cards, for an attack total of just 5.

Bumblebee, Courageous Scout.  Bumblebee is a better option than Slipstream, as his base attack already starts from 6. In a deck filled with white icons, his ability is very often equivalent to Bold 2. Its ability also triggers off defensive white flips. Therefore, it also simulates Tough 2, which is not totally irrelevant in a deck with 9 blue icons. His chances of attacking  for less than 10 are less than 5%.

Arcee, Skilled Fighter. Arcee doesn't have any specific synergy with the compression cannon. Her synergy his directly with knocking out your enemies, which makes her worth considering.  

Starscream, Decepticon King. Starscream is not considered here because of its synergy with the cannon, but because he can actually make good use of some 🅆🄶 cards, namely Decepticon Crown and Secret Dealings. These are cards we would include in our deck anyway (with no doubt about the crown). Besides, Starscream's high base attack makes mixed color builds preferable to almost mono-color ones. The distribution of his attack total is not plotted, given how much it depends on the number of crowns we flip. But the compression cannon makes flipping just one crown as good, or even better, than flipping more than one. Planning for a crown means that we're flipping four cards. Two colors come from the crown itself, for an additional +2. The remaining three cards will almost always feature at least another color, but very often two, and sometimes 3.

Wheeljack, Weapons Inventor. With King Starscream we got a first glimpse at ways Blowpipe might be used in competitive scenarios. The other part of the story is gonna be told by Wheeljack. In a deck optimized at flipping 4 or 5 colors off the natural 2 flips, Bold makes flipping all 5 colors very simple (figure 10). In his bot mode, and with a weapon in our scrap pile, Wheeljack has innate Bold 3, meaning that we almost always flip 7 cards off his attack. The most likely attack boost is +7, corresponding to 5 different colors and 2 orange icons (figure 11).

Given how impactful Bold is in increasing the average number of colors we flip, we might consider relaxing the tight constraints we put on our deck. In the second part of these notes we will concede to the allure of aggressive, unsophisticated strategies. We'll just splash non-orange icons, and consider more competitive ways of playing Blowpipe in an orange shell.

Part II - How to Build a 5★ Lion

We can either put a lot of thought into maximizing the attack boost with our compression cannon, or just play orange and Bold. The second option seems to have much better chances of winning us some games. Therefore, it's time to part from our mathematical playground, and consider the role of Blowpipe in an orange shell. The main subject of this section is the comparison between Blowpipe, and the archetypal aggressive Battlemaster, Lionizer. We'll see that Blowpipe can be just as aggressive, with the additional bonus of costing 2★ less. 

Part I was entirely focused on a deck with 22 white cards, which many might rightfully consider debatable at best. Here we consider a color composition we might choose anyway for an aggro deck. The only additional cost is going to be spacing for the suboptimal 🄾🄺 cards: 

• 6 🄾🄶 cards (Enforcement Batons, Bashing Shield, Press the Advantage, are just some of the good examples.)

• 6 🄾🄺 cards (No options in this case. Let's play Rock Toss and Combat Dagger, and stop whining.)

• 3 🄾🄱 cards (Maybe Roll Out!)

• 3 🅆🄶 cards (Spare Parts to protect the cannon, Espionage, Escape Route, etc.)

• 3 🅆 cards (Quartermaster)

• 3 🄱 (Start your Engines, Dino Chomp! Don't forget Swap Missions.)

• 10 🄾 cards

• 6 🄾🄾 cards

Let's consider again a Bold 3 Wheeljack (a double-flipped Grimlock would work as well) first upgraded with the compression cannon (Blowpipe), and then with the RS Firesteel Saber (i.e. Lionizer, Ground Command · Artillery in his "weapon mode"). The striking similarity between the two cases is shown in the two panels of figure 12. As we can see, Lionizer is still slightly more aggressive, but definitely not 2★ more aggressive. 

Our last figure shows the probability distribution of the number of colors we flip. It's worse than the one in figure 10, as it should be, given that figure 10 is the solution of an optimization problem. We are trading a +1 or +2 bonus in figure 13 for the largely increased amount of orange icons we flip. 

As a final remark, another little reward we get from choosing Blowpipe over Lionizer is that we still gain at least an additional +1 bonus -more if we flip more than just orange- even when our opponent shuts down Bold.

External Links

Ken Nagle - Nagle’s Notes: A Colorful Raider