Panoptic Mirror + Precursor Golem + Rite of Replication works for insane implosion?

[GGoober]

Panoptic Mirror imprinting Rite of Replication targeting Precursor Golem every turn should work if I'm correct. Since Kicker costs work on Scepter-Chant combo.

By work I mean having the abillity to cast Rites on my upkeep with Kicker cost targeting Precursor Golem/Golem tokens.

Just wanted to double check. This is going to be my pet casual deck if the interaction work in what I hope it does.

[Nihil Credo]

I don't see any reason why it wouldn't work. You'd have to pay the Kicker cost separately, of course (as with Scepter/Chant).

(Though a single kicked RoR gives you 84 power's worth of Golems (3*3 + 5 * 2*3 + 5 * 3*3), and any additional ones will not make your army much more resilient, so the Panoptic Mirror may just be a little wasted compared to, say, a Leonin Abunas.)

[rufus]

Panoptic Mirror imprinting Rite of Replication targeting Precursor Golem every turn should work if I'm correct. Since Kicker costs work on Scepter-Chant combo.

By work I mean having the abillity to cast Rites on my upkeep with Kicker cost targeting Precursor Golem/Golem tokens.

Just wanted to double check. This is going to be my pet casual deck if the interaction work in what I hope it does.

You'll need to play the kicker cost (it's additional) if you want it, and, really, do you need to cast it more than once when the first time (unkicked) will give you an additional 5 3/3s for a total of 8 golems.

It's rather silly. You could just throw Might of Old Krosa or something similar on the golems, and swing for the win.

[GGoober]

Yeah Nihil, I was interested in RoR + Precursor Golem after reading up some posts and finding it absurd how it crashes Magic Online games. I've written out a detailed mathematical proof/formul outlining the governing relationship between Rites, Precursor Golem and Tokens. The number of tokens/PGolems you get is a function of the starting population of tokens/PGolems, and whether you choose to rite the tokens/PGolems.

But when you get down to the third kicked Rites, I believe it gets VERY out of hand. So I'm basically bored, and want to make a deck with Panoptic Mirror + Precursor Golem + Rite of Replication + Doubling Season, and tabulate a relationship on how many Golem/Tokens I get based on the starting population and just crack up in casual games :P


The proof's below, people can read if they're interested :P (Obviously it's hard to define a well-defined function between mathematical functions and magic rules, so I've done my best to come up with the best functional/mathematical representation of spells/magic rules/cards)

For those who have some 'rigorous' math background or an interest in mathematical proofs, I've written up a proof/formula computing the interactions of Rite of Replication and Precursor Golem.

It's a funny interaction, but a lot of subtle things are incorrectly assumed, e.g. you get different number of tokens if you kicked Rites on golem tokens instead of Precursor Golems. The situation becomes more complicated if you don't just focus on g + 2t (g = Precursor Golem, 2t = 2 Golem Tokens) populations. The below proof details a general formula to deduce how many Precursor Golem + Token population you get for a given initial starting population of Precursor Golem + Tokens.

The 2 formula guarding the rate of Golem reproduction with Rites of Replication is:


S(g) = aS(C((a-1)g,bt)) + R(g)
S(t) = aS(C(ag,(b-1)t)) + R(t)

where
g = Precursor Golem
t = Golem token
S(g1,g2,g3,...) = the act of casting a sorcery/instant spell (e.g. Rite of Replication) choosing legal targets g1,g2,g3,...
R(g) = 5G(g) = Rite of Replication resolving (kicked) which puts 5 copies of g, where G(g) accounts for all ETB effects of g.
C(g,t) to be the collection of Precursor Golem and Golem tokens.



For further details of the proof/logic follow below, but you might get lost because I've lost myself trying to be as accurate as I can :P











DEFINITION:
Let the transformation S->R be defined as a spell (S) resolving leading its effect (R). (note that if S is countered, then this transformation does not exist).
Define G(g) to be the enter-the-battlefield (ETB) effects of object g.


DEFINE:
g = Precursor Golem
t = Golem token
S(g1,g2,g3,...) = the act of casting a sorcery/instant spell (e.g. Rite of Replication) choosing legal targets g1,g2,g3,...
G(g) = g+2t = triggered effect of Precursor Golem when it ETB.
R(g) = 5G(g) = Rite of Replication resolving (kicked) which puts 5 copies of g, where G(g) accounts for all ETB effects of g.
C(g,t) to be the collection of Precursor Golem and Golem tokens.
N(g,t) = S(g) + C(g,t) to be the resulting collection of Precursor Golem and Golem tokens (including the initial collection C(g,t)) after S is cast on g.


PROPERTIES OF 'g':
Assume a collection C(ag,bt) of (a) Precursor Golems and (b) Golem tokens

Then S(g) = aS(C((a-1)g,bt)) + R(g)
i.e. spell (S) targeting a Precursor Golem will rite of replicate the original Precurso Golem [component R(g) = 5g] and the other (a-1) Precursor Golems triggering Rite of Replication (a) times on (a-1) Precursor Golems and (b) Golem tokens [component aS(C((a-1)g,bt))]

However, S(t) = aS(C(ag,(b-1)t)) + R(t)
i.e. spell (S) targeting a Golem token will rite of replicate the original Golem token [component R(t) = 5t] trigger Rite of Replication (a) times on (a) Precursor Golem and (b-1) Golem tokens [component aS(C(ag,(b-1)t))]

Note how different S(x) behaves under different domain x, i.e. x = g (Precursor Golem) and x = t (Golem token) gives different results.





THEOREM:
R(ax + by) = aR(x) + bR(y) is a linear function.

Proof:
By definition, a linear function satisfies: f(ax+by) = af(x) + bf(y) for all {x,y} and a,b = constants

From the defintion, R(g) = 5g, we have:
R(ax+by) = 5(ax+by) = 5ax + 5by
= a(5x) + b(5y) [commutativity of real numbers]
= aR(x) + bR(y) for all {x,y} and a,b = constants
QED



Having our tools, definitions, and theorems, lets seek to explore the application of the theorems.


1st PROBLEM OF INTEREST:
What happens when you kick a Rite of Replication targeting Precursor Golem with two Golems tokens in play?


In this problem, C(g,t) = g + 2t is our initial condition (but feel free to set 2t to any ct where c is a constant)



Hence applying the above defintions, this problem can be rephrased as:


DECLARE TARGETS FOR RITE OF REPLICATION:
=>N(g,t) = S(g) + C(g,t) [kicking Rite of Replication on Precursor Golem with C(g,t) = g+2t in play]

RESOLVE TRIGGERS ARISING FROM TARGETS OF S:
=> N(g,t) = S(C(0g,2t)) + R(g) + C(g,t) [Using the property: S(g) = aS(C((a-1)g,bt)) + R(g) with initial conditions a=1, b=2]
=> N(g,t) = S(0+2t) + R(g) + [g+2t] [applying the initial condition: C(g,t) = g+2t]
=> N(g,t) = R(2t) R(g) + [g+2t] [resolving spell i.e. applying the transformation S->R]

RESOLVE ENTER-THE-BATTLEFIELD TRIGGERS:
=> N(g,t) = 2[5t] + 5G(g) + [g+2t] [applying definition of R(g) = 5G(g), note that R(t)=5t i.e. T(t)=t since tokens do not have ETB triggers]
=> N(g,t) = 10t + 5(g+2t) + g + 2t [applying definition G(g) = g+2t]
=> N(g,t) = 10t + 5g + 10t + g + 2t
=> N(g,t) = 6g + 22t



That was easy right? But without a formulistic way of approaching this problem, can you solve a slightly harder problem?




2nd PROBLEM OF INTEREST:
What happens when you kick a Rite of Replication twice targeting the same Precursor Golem with two Golem tokens in play?

Have fun working it out! But here's my formulistic solution based on the above:


Applying the result of the first Rite of replication, we have the result: S(g) -> 6g + 22t from above
Now apply S(g) on the collection C(g,t) = 6g + 22t


DECLARE TARGETS FOR RITE OF REPLICATION:
=> N(g,t) = S(g) + C(g,t) [kicking Rite of Replication on Precursor Golem with C(g,t) = 6g+22t in play]

RESOLVE TRIGGERS ARISING FROM TARGETS OF S:
=> N(g,t) = 6S(C((6-1)g,22t)) + R(g) + C(g,t) [Using the property: S(g) = aS(C((a-1)g,bt)) + R(g) with initial conditions a=6, b=22]
=> N(g,t) = 6S(5g+22t) + R(g) + [6g+22t] [applying the initial condition: C(g,t) = 6g+22t]
=> N(g,t) = 6R(5g+22t) + R(g) + [6g+22t] [resolving spell i.e. applying the transformation S->R]
=> N(g,t) = 30R(g) + 132R(t) + R(g) + [6g+22t] [applying linearity of R]

RESOLVE ENTER-THE-BATTLEFIELD TRIGGERS:
=> N(g,t) = 150G(g) + 132[5t] + 5G(g) + [6g+22t] [applying definition of R(g) = 5G(g), note that R(t)=5t i.e. T(t)=t since tokens do not have ETB triggers]
=> N(g,t) = 150[(g+2t)] + 660t + 5(g+2t) + 6g + 22t [applying definition G(g) = g+2t]
=> N(g,t) = 150g + 300t + 660t + 5g + 10t + 6g + 22t
=> N(g,t) = 161g + 992t



3rd PROBLEM OF INTEREST:
What happens when you kick a Rite of Replication twice targeting first the Precursor Golem then a Golem Token in play?

This is slightly non-intuitive that the answer will be different from the 2nd PROBLEM OF INTEREST, but the subtlety is given by how S(x) is defined on the domain x, i.e.

S(g) = aS(C((a-1)g,bt)) + R(g)
S(t) = aS(C(ag,(b-1)t)) + R(t)

Nevertheless, recognizing the subtle difference means having the ability to solve this problem!


DECLARE TARGETS FOR RITE OF REPLICATION:
=> N(g,t) = S(g) + C(g,t) [kicking Rite of Replication on Precursor Golem with C(g,t) = 6g+22t in play]

RESOLVE TRIGGERS ARISING FROM TARGETS OF S:
=> N(g,t) = 6S(C(6g,21t) + R(t) + C(g,t) [Using the property: S(t) = aS(C(ag,(b-1)t)) + R(t) with initial conditions a=6, b=22]
=> N(g,t) = 6S(6g+21t) + R(t) + [6g+22t] [applying the initial condition: C(g,t) = 6g+22t]
=> N(g,t) = 6R(6g+21t) + R(t) + [6g+22t] [resolving spell i.e. applying the transformation S->R]
=> N(g,t) = 36R(g) + 126R(t) + R(t) + [6g+22t] [applying linearity of R]

RESOLVE ENTER-THE-BATTLEFIELD TRIGGERS:
=> N(g,t) = 180G(g) + 127[5t] + [6g+22t] [applying definition of R(g) = 5G(g), note that R(t)=5t i.e. T(t)=t since tokens do not have ETB triggers]
=> N(g,t) = 180[(g+2t)] + 635t + 6g + 22t [applying definition G(g) = g+2t]
=> N(g,t) = 180g + 360t + 635t + 6g + 22t
=> N(g,t) = 186g + 1017t



Therefore if you want to make more Golem/tokens, Rite of Replicate your tokens, and not the Precursor Golem! (although this may not be true for some combination of a,b in C(ag,bt) but I don't have time for a proof/corollory for this lol)

QED Biatch

[Nihil Credo]

A smarter formalisation would have been to use two different functions, S_t for casting RoR on a token and S_g for casting RoR on a golem, using the current creature total as the argument. Also you can just incorporate ETB effects within R from the start and skip that silly business with its linearity, which is trivial since you define R using linear operations.

[a, b are natural numbers; (a, b) is a vector where a is the number of golems and b that of tokens]

R(a, b) = (5a, 5*(2a+b))

S_t(a, b) = (a, b) + R(0, 1) + aR(a, (b-1)) = (a + 5 a^2, 5 + b + 5 a (-1 + 2 a + b))
S_g(a, b) = (a, b) + R(1, 0) + aR((a-1), b) = (5 + a + 5 (-1 + a) a, 10 + b + 5 a (2 (-1 + a) + b))

those two last expansions courtesy of Mathematica because I couldn't be arsed. Now that you have explicit formulas on a vector space you can have much more fun, more easily, e.g. defining and examining (S_x)^n.

[GGoober]

Thanks Nihil, that is win, making it much more concise and elegant. Instead of a single-variable function, why not just use multivariable. (how did I not see/remember that? herp lol). And I'm remembering a little bit of my college maths lol. Going to work on putting your equations in matrix form, since it's just two linear-equations. I'll get that done when I'm home tonight and post it here :D Better post it on another board. This one is for rules questions and the thread has been off-topic enough. ~NC

And off to work on the deck now that this works as intended! I doubt I can order enough Golem tokens and Precursor Golems to support the deck though.

[Jason]

OK. I need a clarification. So Rite of Replication with the kicker on a Precursor Golem will result in 6 Precursor Golems and 22 Golem tokens. That's fine. But if Rite of Replication with the kicker is cast on a Precursor Golem again, the Rite of Replication isn't hitting each Golem 5 times, is it? The question boils down to this text:
"Whenever a player casts an instant or sorcery spell that targets only a single Golem, that player copies that spell for each other Golem that spell could target. Each copy targets a different one of those Golems."
My question: this does not generate multiple triggers with multiple Precursor Golems, correct?

(A simpler scenario: 2x Precursor Golems, some number of Golem tokens. Lightning Bolt targeting a Golem isn't going to hit each Golem twice?)

[majikal]

OK. I need a clarification. So Rite of Replication with the kicker on a Precursor Golem will result in 6 Precursor Golems and 22 Golem tokens. That's fine. But if Rite of Replication with the kicker is cast on a Precursor Golem again, the Rite of Replication isn't hitting each Golem 5 times, is it? The question boils down to this text:
"Whenever a player casts an instant or sorcery spell that targets only a single Golem, that player copies that spell for each other Golem that spell could target. Each copy targets a different one of those Golems."
My question: this does not generate multiple triggers with multiple Precursor Golems, correct?

(A simpler scenario: 2x Precursor Golems, some number of Golem tokens. Lightning Bolt targeting a Golem isn't going to hit each Golem twice?)
It would appear that it does, indeed, trigger for each Precursor Golem you have in play, for the same reason multiple Riddlesmiths trigger when you play artifact spells.

This is also why you want to target your tokens instead of your Precursor Golem for each Rite beyond the first, because you'll get more copies.

[rufus]

(A simpler scenario: 2x Precursor Golems, some number of Golem tokens. Lightning Bolt targeting a Golem isn't going to hit each Golem twice?)
The original target is only hit by the orignal bolt. All others are hit twice.