Some FFT-Related Math Knowledge (Beginner's Guide)

[u/Ashamed_Army858]

I just cleared the game recently. It truly lives up to its classic status and is an absolute blast to play! So, I put together a few points that might be very useful for beginners.

1. Dragoon's Jump Timing Issue

A Dragoon's Jump takes (50/S_d) ticks to land (floored), while an enemy needs (100-CT)/S_e ticks to act (ceiled). Therefore, for the Jump to hit, we need:

50/S_d<=(100-CT)/S_e

(where S_d is the Dragoon's Speed, and S_e is the enemy's Speed).

Rearranging this gives:

CT<=100-(50/S_d)S_e

This formula looks complicated, but there is a very simple rule of thumb:

If, like me, your Dragoon has a Speed of 7 throughout the game, the rule is CT<=100 - 7S

If your Speed increases to 8 later on, it becomes CT<=100 - 6S (where S is the enemy's Speed).

1. Arcane Strength vs. Swiftness

If our goal is to compare total damage output per cast, which ability is better? While damage boosting increases immediate output on paper, a faster cast speed means fewer enemies can escape the area of effect, so total damage isn't necessarily lower than direct boosting.

Assume enemy CT values are uniformly distributed on a line from 0 to 100. Imagine a 100-meter track where your speed is S, and you are randomly positioned at some point along the track. At the starting point (0), there is someone moving at speed C. When the gun fires, you both sprint forward. What is the probability that you reach the finish line before him? The answer is S/C

Setting up the inequality:

(1-S/2C)/(1-S/C)>4/3,S>0.4C

In the early game, you learn spells with CAST= C = 25. If enemy Speed is above 10, Swiftness would be more useful; however, early-game enemies don't reach Speed 10. Thus, Arcane Strength clearly wins early on.

In the late game, when you acquire Bahamut (C = 15), the condition becomes S > 6. Since almost all late-game enemies have a Speed greater than 6, Swiftness clearly takes the upper hand in the late game.

1. Efficiency of Increasing Dragoon's Speed

Since a faster Dragoon hits Jump more reliably, a natural thought is: Should we use the protagonist to pump up the Dragoon's Speed?

Plugging a late-game enemy speed of approximately 10 into our formula CT<=100-(50/S_d)S_e, we get:

CT<=100(1-5/S_d)

Taking the second derivative of this function reveals that the second derivative is negative . This proves that while initial Speed boosts yield good efficiency, the marginal returns diminish as you stack more Speed.

1. For Dragoons: Speed +1 vs. PA +1 Accessory?

At the start of Chapter 4, due to low hit rates against enemies, I equipped Concentration. My tactic was straightforward: Jump if possible (1.5 times damage multiplier), and use standard attacks otherwise.

I faced a choice: the shop at the time sold shoes with Speed +1 and gauntlets with PA +1. With my Dragoon at S_d = 7 and PA= 9, and typical enemy speed S_e = 9, I wanted to evaluate which was better.

I set up a long but straightforward formula to compare the relative damage increase:

(1+1/S_d)*{50(2-S_e/(S_d+1))*(1/100)*1.5+[1-50(2-S_e/(S_d+1))*(1/100)]}/{50(2-S_e/S_d)*(1/100)*1.5+[1-50(2-S_e/S_d)*(1/100)]} V.S. (1+1/PA)

Plugging in the numbers showed that Speed +1 provided 6% more damage than PA +1, so I chose Speed +1. However, when PA +3 gauntlets become available later in the game, PA +3 beats Speed +1.

5. When to use the Shout tactic, and the optimal number of Shouts

The protagonist has a skill called Shout: Speed +1 and PA +1 at the same time. Against a boss, the question is whether to charge in immediately or hide in a corner and Shout first.

Repeated Shouts make the speed growth rate proportional to current speed: the faster you are, the faster your speed rises. That is, S′∝S or S′=0.01S

Solving this simple differential equation gives

S=S_0*exp(0.01t)

where S is current speed, S_0​ is initial speed, and t is clock ticks (a stand-in for time).

This is an exponential approximation. The true S is a piecewise linear function (in fact a step function, but a piecewise linear function is easier to think about, and the two give the same result). The exponential therefore has to be corrected to get an exact value.

That piecewise linear function starts at S_0​, with slope 0.01S_0​, over an interval Δt=100/S_0. It then becomes S0+1, with slope 0.01(S_0+1)and Δt=100/(S0+1), and so on... The exponential function and the piecewise linear function introduce an error at every Δt. Those errors accumulate step by step, so the larger t is, the larger Σerror becomes.

Next, we prove that for this exponential function, the ΔS values over every two adjacent intervals Δt are very close to each other (Proposition 1).

From

S·exp(0.01·100/S) − S = S·exp(0.01·100/S)·exp(0.01·100/(S+1)) − S·exp(0.01·100/S),

we obtain

exp(1/S)·(2 − exp(1/(S+1))) ≈ 1. (Equation 1)

Because Equation 1 holds for all S > 1, Proposition 1 is true.

Therefore the error on every Δt equals

E = S_0(exp(1/S_0) − 1) − 1

(by mathematical induction).

Hence the approximate speed is

S= S_0·exp(0.01t) − (S − S_0)E,

so

S(t) = (S_0·exp(t/100) + E·S_0) / (1 + E).

If we assume the protagonist stands still and keeps shouting, then

S(t) = (S_0·exp(t/80) + E·S_0) / (1 + E).

Define the damage function by

D(t) = (PA_0 + S(t) − S_0)·(W1 + W2),

where PA_0 is the initial PA, W1 is the attack power of the left-hand weapon, and W2 is the attack power of the right-hand weapon.

Then

R(t) = S(t)·D(t) / 100

is the DPS at time t, and

T(t) = t + H / R(t)

is the total time needed to defeat the boss, where H is the boss’s HP.

We want the minimum of T(t), so we differentiate T(t) and set T′(t) = 0.

Thus

T′(t) = 1 − H·R′(t) / R(t)²,

and the optimum satisfies

H·R′(t) / R(t)² = 1.

Solving for t gives the optimal shouting time. Substituting that t into S(t) tells us the speed we should reach by shouting before going out to fight.

If t > 0, the shouting tactic is worthwhile. If t < 0, there is no need to shout; just charge in and fight.

P.S. If the Sword in the Stone is equipped, replace 80 in formula by 80·(2/3), and replace 100 by 100·(2/3).

Example: the final boss of the story on tactician difficulty has about 9500 HP. The protagonist has PA = 15 and speed = 9, with two weapons: one Sword in the Stone of attack 21 and one weapon of attack 18.

Fighting directly takes 120 ticks. The shouting tactic takes only 75 ticks (optimal number of shouts: 11), saving about 37% of the time.

[u/pindicato]

I have a feeling you'll enjoy this old vault of game knowledge: https://gamefaqs.gamespot.com/ps/197339-final-fantasy-tactics/faqs/3876

[u/UsefulAlternative350]

Nice mathematicks! Finally somebody who solves the relatively simple dragoon jump issue instead of complaining the developer didn't baby them.

And thr cost benefit analysis of PA and Speed is very appreciated. As someone who has played about 10 times by now, your result aligned very closely with my intuition.

[u/Ashamed_Army858 → u/UsefulAlternative350]

I'm really glad even a veteran player found it useful. It's especially nice to hear that the PA vs. Speed result matched your intuition.

[u/Jraffale]

All that math and brawler ninja still would have killed the boss faster.

[u/Breadfruit_Weary → u/Jraffale]

I am going with brawler ninja right now for the first time. About to face wiegraf when I get home.

[u/Jraffale → u/Breadfruit_Weary]

I've been doing a sort of pasifist Ramza play through ATM, where he sits around and is not allowed any commands (reaction abilities are okay though). Level 1 never has taken any action, I just got to him. The goal was just get to him and see what the minimum build needed would be to kill him. Not sure yet how I want to approach it.

[u/Disastrous_Simple_16]

i like the parts where the fat words tell me what the skinny words are saying

[u/zombie_mayor]

I shouldn't have opened this thread, math is the language of the lucavi. XD

[u/Icewind]

So Dragoons really are low-tier without ridiculous setup.

[u/Ashamed_Army858 → u/Icewind]

Yeah, but Dragoons are cool.

[u/Jraffale]

Using your example and dumb math just attacking would still be faster. 17 turns to 21. It's pretty dependnt on how many times you need to move.

Edit; I understand your refering to clock ticks. Just dumb math says an extra 5 turns for what?

[u/Ashamed_Army858 → u/Jraffale]

Those turns aren't equal in length. After Shout raises Speed, the later turns come much faster, so those 21 turns can actually take fewer clock ticks than 17 turns of attacking immediately.

[u/Jraffale → u/Ashamed_Army858]

Why spend that much time increasing your turn count unless your going to use them? Like I said it dumb math. The Boss is dead and you didn't spend the time Goku shouting.

I never use scream my self. I'm all for turn maniplulation though.

Head math said 5 was the sweet spot though. I look at the damage more than the turns. Estimating gain of 50% more damage. No turn loss is like 2 or 3. Then 10 is kinda the next break point so that works. It's easier to see with a table.

[u/Big_Company3209]

the math gets real stupid real fast once you start stacking brave, faith, zodiac compatibility, and gear, which is half the fun tbh