Psionic Equations
This page is an in-depth exploration of equations used for psionics.
Conventions:
- PST = Psi Strength
- PSK = Psi Skill
- MC = Mind Control
- Values such as "95/16" represent a unit with Psi Strength 95, Psi Skill 16.
Psionic Equations
The equations governing psionics are as follows:
Attack Strength (AS) = INT( Psi Strength * Psi Skill / 50 )
Defense Strength (DS) = INT( Psi Strength + ( Psi Skill/5 ) )
Attack Chance (A%) = 100/56 * ( Constant + AS - DS - Distance )
where
Constant = 25 for Mind Control
45 for Panic
For this equation, Attack Chance (A%) is always a number (0 to 100), not an actual percent (0.00 to 1.00). Values less than 0 mean guaranteed failure, and greater than 100 mean guaranteed success.
The further away a target is from its attacker, the lower the chance an attack will succeed. Distance follows the "walking TUs" method, where a distance 10 tiles directly lateral equals 10 in the equation, but diagonal distances are more. For example, 10 tiles away on a pure diagonal is considered to be 15 tiles away; see this for more on Walking TUs distance. The distance factor means that beyond a certain point, aliens can only panic a unit, not mind-control it.
Things that don't matter:
- Having Line of sight (direct visualization) on the target
- Aliens use the same equation as soldiers
- Different X-COM editions (DOS 1.4 also WinCE Gold) also doesn't matter
- Differences in elevation have no effect; this doesn't add to distance. (Careful, though - this means you're at distance 0 when directly over or under aliens - see Aliens Vs. X-COM, below.)
The Attack Chance equation can be re-arranged to compute particular values:
Attack Strength
When re-arranged to have AS on the left, the equation becomes:
AS = DS + .56A% - Constant + Distance AS for 100% MC = DS + 56 - 25 + Distance = DS + Distance + 31 AS for 0% MC = DS + 0 - 25 + Distance = DS + Distance - 25
For Panic percent success instead of MC, subtract 20 from the right-hand side of equation.
Defense Strength
DS = AS - .56A% + Constant - Distance DS for 0% MC = AS - 0 + 25 - Distance = AS - Distance + 25 - never MCed DS for 100% MC = AS - 56 + 25 - Distance = AS - Distance - 31 - always MCed
For Panic instead of MC, add 20.
Distance
The distance at which a particular effect can be accomplished can also be computed. Think of the attacker as surrounded by two concentric circles where anything within the inner circle (of distance a.k.a. radius R) will have 100% attack success, and anything outside the outer circle will have 0% success.
Distance = AS - DS - .56A% + Constant Distance for 100% MC = AS - DS - 56 + 25 = AS - DS - 31 Distance for 0% MC = AS - DS - 0 + 25 = AS - DS + 25
For Panic instead of MC, subtract 20. (The "concentric rings" for Panic success are 20 tiles closer to the attacker.)
As you may have guessed, the outer distance/radius is R+56. In other words, the outer, 0% circle is 56 tiles radius farther than the inner circle of 100%. Compare that a 5x5 tileset map is only 75 tiles across at its widest diagional... there's a huge area within the inner and outer circles!
Psi Stats versus Psi Strengths
As shown in the equation, psi strength (PST) and psi skill (PSK) do not contribute equally to Defense Strength (DS) and Attack Strength (AS).
DS is PST plus a fifth of your PSK. Basically, soldiers are born with most of their DS, and can only add 20 more points by maxxing their PSK.
AS is PST*PSK/50. Because PST doesn't change for soldiers, another way to consider the AS equation is a PST/50 "view". PST/50 can vary from 0 to 2 (for PST 0 to 100), which then multiplies your PSK. So if your soldier has a middling PST (e.g. 50), each new PSK point adds one more point to his AS. If she's a psi master (PST 100), each new PSK point adds 2 more to her AS. Because your PSK can increase up to 6 points per combat, a psi master's AS can increase up to 12 points per combat.
Tables for DS and AS, by PST and PSK:
Defense Strength
PSK
0 10 20 30 40 50 60 70 80 90 100
PST 0 0 2 4 6 8 10 12 14 16 18 20
10 10 12 14 16 18 20 22 24 26 28 30
20 20 22 24 26 28 30 32 34 36 38 40
30 30 32 34 36 38 40 42 44 46 48 50
40 40 42 44 46 48 50 52 54 56 58 60
50 50 52 54 56 58 60 62 64 66 68 70
60 60 62 64 66 68 70 72 74 76 78 80
70 70 72 74 76 78 80 82 84 86 88 90
80 80 82 84 86 88 90 92 94 96 98 100
90 90 92 94 96 98 100 102 104 106 108 110
100 100 102 104 106 108 110 112 114 116 118 120
Attack Strength
PSK
0 10 20 30 40 50 60 70 80 90 100
PST 0 0 0 0 0 0 0 0 0 0 0 0
10 0 2 4 6 8 10 12 14 16 18 20
20 0 4 8 12 16 20 24 28 32 36 40
30 0 6 12 18 24 30 36 42 48 54 60
40 0 8 16 24 32 40 48 56 64 72 80
50 0 10 20 30 40 50 60 70 80 90 100
60 0 12 24 36 48 60 72 84 96 108 120
70 0 14 28 42 56 70 84 98 112 126 140
80 0 16 32 48 64 80 96 112 128 144 160
90 0 18 36 54 72 90 108 126 144 162 180
100 0 20 40 60 80 100 120 140 160 180 200
Attack Strength is symmetrical, but Defense Strength is not.
Alien Psi Stats
Alien psi stats for all levels have been summarized here. For a spreadsheet of all psi stats (only) see this, or for ALL alien stats, see this.
Most players (these days!) are playing at Superhuman difficulty level. Here's a table of all Superhuman psi stats, sorted by decreasing Defense Strength:
Alien/Rank DS AS PST PSK N Cyberdisc 116 - 116 - 1 Sectopod 116 - 116 - 1 Silacoid 92 - 92 - 1 Ethereal Commander 86 87 75 58 1 Zombie 80 - 80 - 1 Ethereal Leader 79 71 69 52 1 Sectoid Commander 71 80 58 69 1 Sectoid Leader 69 67 58 58 1 Celatid 69 - 69 - 1 Ethereal Soldier 67 53 58 46 1 Chryssalid 58 - 58 - 1 Snakeman Commander 58 - 58 - 1 Floater Commander 52 - 52 - 1 Snakeman Leader 52 - 52 - 1 Floater Leader 46 - 46 - 1 Sectoid (Common) 46 - 46 - 4 Snakeman (Common) 46 - 46 - 3 Floater (Common) 40 - 40 - 4 Reaper 40 - 40 - 1 Muton (Common) 29 - 29 - 3 Civilian (Common) 5 - 5 - 2
As for Defense Strength:
- The strongest defenders are robots (DS 116), then exotics (silacoid 92, zombie 80) and psi-active aliens (86-67).
- Squarely in the middle are celatids (69), chryssalids (58), and psi-inactive leaders (58-46 for Snakeman and Floater LDRs and CDRs).
- Reapers (40) are by far the most susceptible terrorist.
- Common ranks (navigators, engineers, medics, and soldiers except Ethereals) are at the low end (46, 40).
- All mutons are the lowest of any enemy (all 29; mutons do not have leaders).
- Civilians basically have no defense at all. Ironically, this might mean they'll survive the worst attackers of all better (Ethereals), since many might be MCed! Has anybody seen this?
Here are Superhuman psi stats for psi-active aliens, sorted by descending Attack Strength:
Alien/Rank DS AS PST PSK Ethereal Commander 86 87 75 58 Sectoid Commander 71 80 58 69 Ethereal Leader 79 71 69 52 Sectoid Leader 69 67 58 58 Ethereal Soldier 67 53 58 46
Notice that four out of five have a DS approx. equal to or greater than their AS, but the Sectoid Commander has a lower Defense than Attack strength.
X-COM Vs. Aliens
For every alien at a given distance, a surface (3D) graph of the PST and PSK needed to psionically attack can be made (see inset). But due to the fact that Psi Strength never changes for a given soldier, it's probably easier to consider PST as fixed, in which case only Psi Skill varies - which makes for a simple line equation (Psi Skill vs. Attack Strength, where AS equals percent Panic or MC success).
Along these lines, a table can be made that shows the Psi Skill that a soldier with high Psi Strength (PST 90) needs for a guaranteed (100% AS) chance to Mind Control an alien. Because Psi Strength is randomly determined for recruits, 10% of your soldiers will have PST 90 or higher.
I've shown Superhuman in the table because many of you are probably playing at that level, but the lowest target (Mutons) at Beginner difficulty level is also shown for comparison. If anybody wants to see numbers for other difficulty levels, let me know - MTR.
Some observations can be made, which reflect how the equation operates:
- Distance is always a simple additive. Your Attack Strength must be +15 if a target is at distance 25 instead of 10.
- The difference between 0% chance and 100% chance is always +56 AS. This is due to the slope of the equation (100/56). Here, the equation is re-arranged for an inverted 56/100, which is then multiplied by 100 for the 100% chance.
- Related to this, the Psi Skill needed by the PST 90 soldier in the last column of the table does not rise as fast as defense strength. This is because, at PST 90, every 1 point of PSK increase is worth a 1.8 point increase in Attack Strength (AS= 90/50*PSK= 1.8*PSK). A difference of 56 AS for a PST 90 soldier equals a PSK difference of 31, not 56 (31= 56/1.8= 56/90/50).
- Notice that the psi equation Constant disappears at distance 25 (Constant +25, Distance -25). This shows up as the Defense Strength being equal to the 0% Attack Strength. Maybe the programmers used this "average" distance of 25 as a benchmark to design their equation around.
- Also notice that the weakest possible target - a Beginner Muton at distance 1 - pins down the lower end of where Attack Strength begins to have an effect. If, in theory, you could have a 0 PSK soldier who could psi attack, he would just barely not be able to affect the weakest possible target. Interestingly, if you hack a target to a Defense Strength less than 25, soldiers with the weakest possible Attack Strength (1) can successfully Mind Control them.
- Rookies with a maxxed PST (100) won't quite have the 32 Attack Strength needed to 100% MC an adjacent Beginner Muton right out of the Psi Lab, but they'll be close. 24 is the highest PSK one can have coming out of the lab; 100*24/50 is an AS of 48, which is close to the 57 needed for a 100% MC chance. But then few soldiers will be both 100 PST and 24 PSK right out of the lab (only 1 in 909 - 0.11% - will be), nor is a Beginner Muton always right next to you. :) For a more likely scenario - a 90/20 psi newbie (AS 36), and a Superhuman Muton (DS 29) at distance 25 - you have a 12.5% chance of MC right out of the gate. And you can't touch any of the tougher aliens, not yet - but you can gain 3 to 6 PSK, and 5 to 12 AS (at high PST), each combat.
- The game is well-designed inasmuch as soldiers grow from just barely being entirely psi worthless (unable to affect an adjacent Muton; see above) to being able to MC the toughest targets over long distances, at very high Psi Skill. And you do need high Psi Skill for even high PST soldiers to be certain of MC on the toughest targets. But due to the fact that you may have a number of mid-range PSK soldiers with high PST, each of whom can make 2 or 3 MC attempts, even tough targets are rarely a problem for high PST but mid-range PSK soldiers.
Level Target PST PSK DS 0% AS 100% AS =PSK* Notes
Weakest possible alien target, directly adjacent (distance 1)
Beginner Muton 25 0 25 1 57 32 The weakest possible alien target, directly adjacent
Short range (distance 10)
Beginner Muton 25 0 25 10 66 37 Beginner Mutons
Superhuman Muton 29 0 29 14 70 39 Superhuman Mutons
" Common 46 0 46 31 87 49 Gets all sub-LDR Floaters, Snakes, Sectoids, & Reaper
" Sectoid CDR 58 69 71 56 112 63 Gets all but Ethereal LDR+, Zombie, Silacoid, Robots
" Ethereal LDR 69 52 79 64 120 67 The Zombie is 1 DS higher (DS 80)
" Ethereal CDR 75 58 86 71 127 71
" Silacoid 92 0 92 77 133 74
" Cyberdisc & Sectopod 116 0 116 101 157 88 Robots have the highest DS
All the above, at medium range (distance 25)
Beginner Muton 25 0 25 25 81 45
Superhuman Muton 29 0 29 29 85 48
" Common 46 0 46 46 102 57
" Sectoid CDR 58 69 71 71 127 71
" Ethereal LDR 69 52 79 79 135 75
" Ethereal CDR 75 58 86 86 142 79
" Silacoid 92 0 92 92 148 83
" Cyberdisc & Sectopod 116 0 116 116 172 96
All the above, at long range (distance 50)
Beginner Muton 25 0 25 50 106 59
Superhuman Muton 29 0 29 54 110 62
" Common 46 0 46 71 127 71
" Sectoid CDR 58 69 71 96 152 85
" Ethereal LDR 69 52 79 104 160 89
" Ethereal CDR 75 58 86 111 167 93
" Silacoid 92 0 92 117 173 97
" Cyberdisc & Sectopod 116 0 116 141 197 110
Worst cases at extreme distance (opposite corners of 5x5 map; distance 75)
Superhuman Ethereal CDR 75 58 86 136 192 107
" Cyberdisc & Sectopod 116 0 116 166 222 124
* "=PSK" is the Psi Skill needed for 100% MC, for a soldier with PST 90.
Aliens Vs. X-COM
This table shows the 0% MC success distance for X-COM soldiers versus the five psi-active aliens:
Distance for 0% MC success chance,
for X-COM soldier of Defense Strength ...
Alien/Rank PST PSK AS 0 25 50 75 100 Invulnerable DS
Ethereal Commander 75 58 87 112 87 62 37 12 111
Sectoid Commander 58 69 80 105 80 55 30 5 104
Ethereal Leader 69 52 71 96 71 46 21 - 95
Sectoid Leader 58 58 67 92 67 42 17 - 91
Ethereal Soldier 58 46 53 78 53 28 3 - 77
This is the distance at which you are safe from MC. At any closer distance, you're not safe, at a rate of ~2% MC success per tile closer (1.78% per tile; 100/56).
At 56 tiles closer than the distances shown, you are 100% vulnerable. Thus a total wimp DS 0 soldier can always be MCed by an Ethereal Commander at distance of 56 or less (112-56), that is, over the majority of the map. And of course a 5x5 map is 75 tiles wide at the diagonal, meaning an Ethereal Commander has a good chance of MCing a psi wimp anywhere on any map - as we've all learned to our chagrin.
The Invulnerable column shows the Defense Strength a soldier needs to be entirely safe from MC for each type of alien (a.k.a., 0% MC at distance 1). A DS 100 soldier only just barely has a chance of being MCed even when adjacent to psi active Commanders, and a DS 120 soldier (100/100) is entirely safe from MC, even right next to an Ethereal Commander.
For 0% Panic success distances, add 20 to the distances shown, and also to the Invulnerable column. So a 100/100 (DS 120) soldier can still be panicked by an Ethereal Commander when very near, although the chance is low. You'd have to be DS 131 to be entirely safe from Panic when directly next to an Ethereal Commander, but that would require, e.g., 100/155!
Additional notes:
- The DS 0 to DS 100 columns reflect the distribution of Defense Strength for psi-locked soldiers.
- Elevation does not affect psionics. On a number of UFOs (Abductor, Harvester, and Supply Ship), you are very "close" to aliens on the bridge as you enter the ship. Indeed, if you are directly under them, you are at distance 0, even if they're on the third floor. None of the information on this page directly addresses distances of 0... you'd actually need a DS of 112 to be invulnerable to MC if directly under (or over) an Ethereal Commander.
- In the early days - before you get psi - only about 10% of your crew will be invulnerable to MC by Sectoid Leaders. So if you have 20 soldiers, only about 2 will never lose their wits. Still, it takes a DS 40 to be 100% vulnerable... over half your crew will be better than this. Furthermore...
- Psi-active aliens like to waste their TUs on your wimpiest soldiers, leaving the others free to act.
- But do remember that this changes as a function of distance... that's why things can get really crazy when, e.g., entering a Sectoid Abductor... your Panicked and MCed soldiers may be far afield right at the point that you're higher-DS soldiers enter the ship under its bridge, and need to clear that inner, lower room that's often packed with aliens. Now they may be the "best" targets! (Try keep your weaponless psi wimps close to psi-active aliens, so that the wimps will remain the best targets.)
While all of this is true, don't forget that you typically have many decent psi-unlocked soldiers pretty soon after getting psi, each of whom will get 2 or 3 psi attempts. Combined with the fact that aliens will be going for your wimpiest dudes, chances are very good that you can successfully psi attack even Ethereal Commanders before they successfully attack all of your guys, as we've all seen.
The OSG Speaks
The text of the Official Strategy Guide section on Psionics (p. 227) is reproduced here, for comparison. Thanks to Zombie for transcribing this:
- Chances for Success in Psionic Combat
- Only soldiers with high ratings in psionic strength and psionic skill should use this form of combat, because it is difficult to master. A soldier's psionic combat strength is determined as follows:
- (Psionic strength) x (psionic skill) = (psionic combat strength)
- A soldier with a psionic strength of 20 and a psionic skill of 50, for example, has a psionic combat strength of 1000.
- When you use a psionic attack on a target, the base chance of success is determined by subtracting the target's psionic defense from the attacker's psionic combat strength. The defense value against mind-control attacks is higher than the defense value against panic attacks.
- The psionic defense against panic attacks is determined as follows:
- (Psionic strength) + (psionic skill/5) + (10) = (panic defense)
- The psionic defense against mind-control attacks is determined as follows:
- (Psionic strength) + (psionic skill/5) + (30) = (mind control defense)
- So the base chance of success for a psionic attack is determined as follows:
- (Attacker's psionic combat strength) - (target's psionic defense) = (base chance)
- For example, a soldier with a psionic combat strength of 1000 attacks an alien with a psionic defense of 950. The base chance of success is 1000 - 950, or 50%.
- Psionic attacks decrease in effectiveness over distance.
- Defending X-COM Soldiers Against Psionic Attack
- When X-COM soldiers fall under alien psionic attack, there is a brief flash around the soldier being attacked. If the attack is successful, the victim either panics or falls victim to alien mind control.
- Psionic defense for X-COM soldiers uses the same formula as shown earlier. It is best to have soldiers with the highest possible psionic abilities because they are much less susceptible to psionic attack. This is especially true late in the game, when more psionically endowed aliens tend to show up.
Taken together, the equations as presented in the OSG would look like this, if re-written in our format:
Attack Strength (AS) = Psi Strength * Psi Skill
Defense Strength (DS) = Psi Strength + ( Psi Skill/5 )
Attack Chance (A%) = Constant + AS - DS
where
Constant = -30 for Mind Control
-10 for Panic
As can be seen, while what the OSG shows is more or less correct in and of itself, it omits important details:
- The Attack Strength portion (PST * PSK /50) doesn't show the divisor of 50
- The slope factor (100/56) is missing
- The distance factor was not shown (1 per tile)
- The constants (OSG 10 and 30 versus actual values of 25 and 45) are 15 points low. Also, as presented (as part of the Defense Strength which is subtracted), they result in negative values which are subtracted, when the constant should be added. This leads to a final variance of -35 for Panic attacks (+25 missing, then -10 more) and -75 for MC (+45 missing, then -30 more), versus the correct results.
The absence of the 50 AS divisor was probably the most egregious omission (a.k.a., "these numbers don't make a bit of sense!"), although the constants being so far off didn't help one bit. One could've gotten a close approximation (results that at least made some sense in the game) if only the slope and distance factors had been omitted. But notice how the OSG author himself (David Ellis) lists an incorrect example - how'd he ever get 950 for a Defense Strength?
All in all, it suggests that Ellis never knew the full equation. This can readily happen when an author talks to developers and they tell him a ton of new, important things, but then later he finds that despite the avalance of information, it still wasn't the whole story... but he doesn't have time to follow up amidst the many other things in the book. Not to put down the OSG... It's a remarkably long and comprehensive game guide, relative to many other single-player game guides. Inevitably, some deep game details don't make it, in the all-important shuffle to get the guide out quickly, so it can hit the streets the same time the game does (the peak buying period).
Be that as it may... due to incompleteness, the OSG's psi equations have long frustrated players attempting to use them.
Note
This is all the work I plan to do for the time being; I think I've covered all the pertinent details. I hope you've enjoyed it. If you'd like something shown in more depth, or think of something else entirely, just let me know! --MTR
See Also
- Psionics
- Psi_Strength
- Psi_Skill
- Field Manual: Psionics
- Psi-Lab
- Psi-Amp
- Mind Shield
- Mind Probe
- Experience
- Experience Training
- Molecular Control
- For background on testing of the psi equations, see MTR_Psi_Testing