## FANDOM

8,892 Pages

This article presents some of the mathematics of Civilization V. It was originally published by alpaca at the CivFanatics forum. Notice that this article represents the state of the game before the December 2010 patch!

Updated on 6/13/2015: Unit Maintenance updated to reflect the actual formula in the DLL - monju125

## Calculating CultureEdit

### Border growthEdit

The cost of a border growth is, with t the number of tiles already claimed

$20 + (10 (t - 1))^{1.1}$

### Formula for Social PoliciesEdit

The first thing we need to know when talking about anything culture-wise is how to calculate it. From the game core XML files and in-game observations, it works like this: You calculate a base cost that depends on the number of policies, then multiply with a factor that depends on your number of cities. As follows:

$n$: The number of cities in your empire

$p_{base} (k)$: The base cost of policy k. Depends only on the number of policies

$p_{cities}(n)$: The cost factor depending on the number of cities in your empire

$p(n,k)$: The total cost of policy k. Depends on both the number of cities and policies already unlocked

The policy cost has to be modified depending on your difficulty setting, game speed and map size. I took the numbers from a post by Yamian. Define these modifiers as:

$m_t$: The game speed modifier (3 for marathon, 1.5 for epic, 1 for normal and 0.67 for quick)

$m_d$: The difficulty coefficient (0.5 for settler, 0.67 for chieftain, 0.85 for warlord, 1 otherwise)

$m_z$: The map size modifier (0.3 for normal size and below, 0.2 for large, 0.15 for huge)

$p_{base}(k) = m_t m_d (25 + (6 k)^{1.7}) = m_t m_d (25 + 21.03k^{1.7})$

$p_{cities}(n) = 1 + m_z (n - 1)$

The policy cost scales linearly with the number of cities and something between linear and quadratic with the number of policies. These factors are combined by multiplication, so the total cost is simply:

$p(n,k) = p_{base}(k) * p_{cities}(n)$ rounded to the next multiple of 5

 k=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 n=1 25 45 90 160 245 345 465 595 745 905 1075 1260 1460 1670 1890 2120 2365 2620 2885 3160 3445 3745 4050 4365 4690 5025 5370 5725 6090 6465 6845 2 30 55 120 205 320 450 605 775 970 1175 1400 1640 1900 2170 2460 2760 3075 3405 3750 4110 4480 4865 5265 5675 6100 6535 6985 7445 7920 8405 8900 3 40 70 145 255 395 555 745 955 1190 1445 1725 2020 2335 2670 3025 3395 3785 4195 4620 5060 5515 5990 6480 6985 7510 8045 8595 9165 9745 10345 10955 4 45 85 175 305 465 660 885 1135 1415 1720 2050 2400 2775 3175 3595 4035 4500 4980 5485 6010 6550 7115 7695 8295 8915 9555 10210 10885 11575 12280 13010 5 55 100 205 350 540 765 1025 1315 1640 1990 2370 2780 3215 3675 4160 4670 5210 5770 6350 6955 7585 8240 8910 9605 10325 11060 11820 12600 13400 14220 15065 6 60 115 230 400 615 870 1165 1495 1865 2265 2695 3160 3655 4175 4730 5310 5920 6555 7215 7905 8620 9360 10125 10920 11730 12570 13435 14320 15230 16160 17115 7 70 125 260 450 690 975 1305 1675 2085 2535 3020 3540 4090 4680 5295 5945 6630 7340 8085 8855 9655 10485 11345 12230 13140 14080 15045 16040 17055 18100 19170 8 75 140 285 495 765 1080 1445 1855 2310 2805 3345 3915 4530 5180 5865 6585 7340 8130 8950 9805 10690 11610 12560 13540 14550 15590 16660 17755 18885 20040 21225 9 85 155 315 545 835 1185 1585 2035 2535 3080 3665 4295 4970 5680 6430 7220 8050 8915 9815 10755 11725 12735 13775 14850 15955 17100 18270 19475 20715 21980 23280 10 90 170 345 595 910 1290 1725 2215 2760 3350 3990 4675 5405 6180 7000 7860 8760 9700 10685 11705 12760 13855 14990 16160 17365 18605 19885 21195 22540 23920 25335 11 100 180 370 640 985 1395 1865 2395 2985 3620 4315 5055 5845 6685 7570 8495 9470 10490 11550 12650 13795 14980 16205 17470 18775 20115 21495 22915 24370 25860 27390 12 105 195 400 690 1060 1500 2005 2575 3205 3895 4635 5435 6285 7185 8135 9135 10180 11275 12415 13600 14830 16105 17420 18780 20180 21625 23110 24630 26195 27800 29445 13 115 210 425 740 1135 1605 2145 2755 3430 4165 4960 5815 6725 7685 8705 9770 10895 12065 13280 14550 15865 17230 18635 20090 21590 23135 24720 26350 28025 29740 31495 14 120 225 455 785 1210 1710 2285 2935 3655 4440 5285 6195 7160 8190 9270 10410 11605 12850 14150 15500 16900 18350 19850 21400 23000 24640 26330 28070 29850 31680 33550 15 130 235 485 835 1280 1815 2425 3115 3880 4710 5610 6575 7600 8690 9840 11045 12315 13635 15015 16450 17935 19475 21070 22710 24405 26150 27945 29790 31680 33620 35605 16 135 250 510 885 1355 1920 2570 3295 4100 4980 5930 6950 8040 9190 10405 11685 13025 14425 15880 17395 18970 20600 22285 24020 25815 27660 29555 31505 33505 35560 37660 17 145 265 540 930 1430 2025 2710 3475 4325 5255 6255 7330 8475 9690 10975 12320 13735 15210 16750 18345 20005 21725 23500 25330 27220 29170 31170 33225 35335 37500 39715 18 150 280 565 980 1505 2130 2850 3655 4550 5525 6580 7710 8915 10195 11540 12960 14445 15995 17615 19295 21040 22845 24715 26645 28630 30675 32780 34945 37165 39440 41770 19 160 290 595 1030 1580 2235 2990 3835 4775 5795 6905 8090 9355 10695 12110 13595 15155 16785 18480 20245 22075 23970 25930 27955 30040 32185 34395 36665 38990 41380 43825 20 165 305 625 1075 1650 2340 3130 4015 5000 6070 7225 8470 9795 11195 12680 14235 15865 17570 19345 21195 23110 25095 27145 29265 31445 33695 36005 38380 40820 43320 45880 21 175 320 650 1125 1725 2445 3270 4195 5220 6340 7550 8850 10230 11700 13245 14870 16575 18360 20215 22145 24145 26220 28360 30575 32855 35205 37620 40100 42645 45255 47930 22 180 335 680 1175 1800 2550 3410 4375 5445 6615 7875 9230 10670 12200 13815 15510 17285 19145 21080 23090 25180 27340 29575 31885 34265 36710 39230 41820 44475 47195 49985 23 190 345 705 1220 1875 2655 3550 4555 5670 6885 8200 9605 11110 12700 14380 16145 18000 19930 21945 24040 26215 28465 30790 33195 35670 38220 40845 43540 46300 49135 52040 24 195 360 735 1270 1950 2760 3690 4735 5895 7155 8520 9985 11550 13200 14950 16785 18710 20720 22815 24990 27250 29590 32010 34505 37080 39730 42455 45255 48130 51075 54095 25 205 375 765 1320 2025 2865 3830 4915 6115 7430 8845 10365 11985 13705 15515 17420 19420 21505 23680 25940 28285 30715 33225 35815 38490 41240 44070 46975 49960 53015 56150 26 210 390 790 1365 2095 2970 3970 5095 6340 7700 9170 10745 12425 14205 16085 18060 20130 22290 24545 26890 29320 31835 34440 37125 39895 42750 45680 48695 51785 54955 58205 27 220 405 820 1415 2170 3070 4110 5275 6565 7970 9495 11125 12865 14705 16650 18695 20840 23080 25410 27835 30355 32960 35655 38435 41305 44255 47295 50410 53615 56895 60260 28 225 415 845 1465 2245 3175 4250 5455 6790 8245 9815 11505 13300 15210 17220 19335 21550 23865 26280 28785 31390 34085 36870 39745 42710 45765 48905 52130 55440 58835 62315 29 235 430 875 1510 2320 3280 4390 5635 7015 8515 10140 11885 13740 15710 17790 19970 22260 24655 27145 29735 32425 35210 38085 41055 44120 47275 50520 53850 57270 60775 64365 30 240 445 905 1560 2395 3385 4530 5815 7235 8790 10465 12265 14180 16210 18355 20610 22970 25440 28010 30685 33460 36330 39300 42370 45530 48785 52130 55570 59095 62715 66420 31 250 460 930 1610 2470 3490 4670 5995 7460 9060 10790 12640 14620 16710 18925 21245 23680 26225 28875 31635 34495 37455 40515 43680 46935 50290 53740 57285 60925 64655 68475 32 255 470 960 1655 2540 3595 4810 6175 7685 9330 11110 13020 15055 17215 19490 21885 24395 27015 29745 32585 35530 38580 41735 44990 48345 51800 55355 59005 62755 66595 70530 33 265 485 985 1705 2615 3700 4950 6355 7910 9605 11435 13400 15495 17715 20060 22520 25105 27800 30610 33530 36565 39700 42950 46300 49755 53310 56965 60725 64580 68535 72585 34 270 500 1015 1755 2690 3805 5090 6535 8130 9875 11760 13780 15935 18215 20625 23160 25815 28585 31475 34480 37600 40825 44165 47610 51160 54820 58580 62445 66410 70475 74640 35 280 515 1045 1800 2765 3910 5230 6715 8355 10145 12085 14160 16370 18720 21195 23795 26525 29375 32345 35430 38635 41950 45380 48920 52570 56325 60190 64160 68235 72415 76695 36 285 525 1070 1850 2840 4015 5370 6895 8580 10420 12405 14540 16810 19220 21760 24435 27235 30160 33210 36380 39670 43075 46595 50230 53980 57835 61805 65880 70065 74355 78745 37 295 540 1100 1900 2910 4120 5510 7075 8805 10690 12730 14920 17250 19720 22330 25070 27945 30950 34075 37330 40705 44195 47810 51540 55385 59345 63415 67600 71890 76295 80800 38 300 555 1125 1945 2985 4225 5650 7255 9030 10960 13055 15295 17690 20225 22895 25710 28655 31735 34940 38275 41735 45320 49025 52850 56795 60855 65030 69320 73720 78235 82855 39 310 570 1155 1995 3060 4330 5790 7435 9250 11235 13380 15675 18125 20725 23465 26345 29365 32520 35810 39225 42770 46445 50240 54160 58200 62365 66640 71035 75545 80170 84910 40 315 580 1185 2045 3135 4435 5930 7615 9475 11505 13700 16055 18565 21225 24035 26985 30075 33310 36675 40175 43805 47570 51455 55470 59610 63870 68255 72755 77375 82110 86965 41 325 595 1210 2090 3210 4540 6070 7795 9700 11780 14025 16435 19005 21725 24600 27620 30790 34095 37540 41125 44840 48690 52675 56780 61020 65380 69865 74475 79205 84050 89020 42 330 610 1240 2140 3285 4645 6215 7975 9925 12050 14350 16815 19440 22230 25170 28260 31500 34880 38410 42075 45875 49815 53890 58095 62425 66890 71480 76195 81030 85990 91075 43 340 625 1265 2190 3355 4750 6355 8155 10145 12320 14670 17195 19880 22730 25735 28895 32210 35670 39275 43025 46910 50940 55105 59405 63835 68400 73090 77910 82860 87930 93130 44 345 635 1295 2235 3430 4855 6495 8335 10370 12595 14995 17575 20320 23230 26305 29535 32920 36455 40140 43970 47945 52065 56320 60715 65245 69905 74705 79630 84685 89870 95180 45 355 650 1325 2285 3505 4960 6635 8515 10595 12865 15320 17955 20760 23735 26870 30170 33630 37240 41005 44920 48980 53185 57535 62025 66650 71415 76315 81350 86515 91810 97235 46 360 665 1350 2335 3580 5065 6775 8695 10820 13135 15645 18330 21195 24235 27440 30810 34340 38030 41875 45870 50015 54310 58750 63335 68060 72925 77930 83065 88340 93750 99290 47 370 680 1380 2380 3655 5170 6915 8875 11045 13410 15965 18710 21635 24735 28005 31445 35050 38815 42740 46820 51050 55435 59965 64645 69465 74435 79540 84785 90170 95690 101345 48 375 695 1405 2430 3725 5275 7055 9055 11265 13680 16290 19090 22075 25235 28575 32085 35760 39605 43605 47770 52085 56560 61180 65955 70875 75940 81150 86505 92000 97630 103400 49 385 705 1435 2480 3800 5380 7195 9235 11490 13955 16615 19470 22515 25740 29145 32720 36470 40390 44475 48715 53120 57680 62395 67265 72285 77450 82765 88225 93825 99570 105455 50 390 720 1465 2525 3875 5485 7335 9415 11715 14225 16940 19850 22950 26240 29710 33360 37185 41175 45340 49665 54155 58805 63615 68575 73690 78960 84375 89940 95655 101510 107510

### How fast do I acquire new policies?Edit

This is very simple, and I'll just list it here to introduce some variable names. It depends on the culture your empire yields. With

$c$: The total culture yield of your empire

$p$: The cost of your next social policy (I will omit the dependencies for readability)

$t$: The time in turns to unlock the next policy

we get the simple formula

$t = p/c$

if we assume we just unlocked a new policy. If we introduce

$c_{acc}$: The culture already accumulated in the culture bucket

we get

$t = (p - c_{acc})/c$

### How do I calculate the total culture in my empire?Edit

This is shown in the UI but again to introduce some concepts. Let

${\bar c} = c/n$ : the average culture of each city in your empire

It makes sense to split c up into a part that depends on n and represents your "typical city culture", the culture each city you newly create will add to your empire, and a part that is constant in n and represents bonuses from wonders, landmarks and city states. This is exact if you immediately buy your typical culture buildings but normally it's an approximation (it represents an equilibrium you will not normally have reached)

$c_t$: The typical culture per city

$c_c$: The city state culture

$c = n c_t + c_c = n {\bar c}$

Before I continue let's look at the policy speed for large numbers

\begin{align} t &= \frac{p_b(k) (1 + p_m (n - 1))}{n {\bar c}} \\ &= \frac{p_b(k)}{\bar c}\left(\frac{1 - pm}{n} + pm\right) \\ \end{align}

$\lim_{n \to \infty}t = \frac{p_m p_b}{\bar c}$

Also: $\lim_{n \to \infty}{\bar c} = c_t$

So for large n, the average amount of culture per city is a good measure for the policy speed.

### Will expanding increase or decrease policy speed?Edit

This is the question Paeanblack and I discussed in said thread. To analyse this, we have to calculate the number of turns to the next policy and see how it's affected by going from n->n+1.

When we do the city number increase, both p and c change. Let p and c be the policy cost and culture yield before founding the new city and p' and c' be the respective numbers after the founding.

$p' = p_b(k) (1 + p_m n) = p + p_m p_b(k)$

$c' = (n + 1) c_t + c_c = c + c_t$

Then calculate t' and check when it gets smaller than t (this would signify an increased policy speed)

$t' = p'/c' = \frac{1 + p_m n}{c_t n + c_t + c_c} < \frac{1 + p_m n - p_m}{c_t n + c_c}$

$(1 + p_m n) (c_t n + c_c) < (1 + p_m n) (c_t n + c_c) + (1 + p_m n) c_t - p_m (c_t n + c_t + c_c)$

$p_m c_t n + p_m c_t + p_m c_c < c_t + p_m n c_t$

$p_m (c_t + c_c) < c_t$

$r_c = \frac{c_c}{c_t} < \frac{1}{p_m} - 1 = \frac{1 - p_m}{p_m}$

For standard-sized maps, $p_m = 3/10$ so

$r_c < \frac{7}{3} = 2.\hat3$

So if the base culture from city states is less than 7/3 times larger than the typical city culture (let's call this the culture ratio r_c), you will get an increased policy speed from expanding. If it's exactly equal to this, the speed will stay the same, and if it's more, policy speed will slow. It should be noted that, no matter the value of r_c, for large numbers of cities the increase or decrease for founding an additional city will be very small.

For larger maps, this will be a little different.

• Large: r_c < 4
• Huge: r_c < 17/3 =5.\hat6

### NumbersEdit

Now we can plug in some example numbers into our calculations. I will discuss the results only for the standard map size

1. Let's assume you only build a monument in each of your cities. This is equivalent to a value of cT = 2. So if cC is five or greater, for example because you have at least one cultural city state as a friend, expansion will slow down your social policy speed.
Standard: cC > 4, Large: cC > 8, Huge: cC > 11
2. Now assume we build a monument and a temple, or cT = 5. Then, cC <= 11 will still yield an increase in your social policy speed. A cC of 10 is still pretty low, though. You normally still get it later on if you have at least one cultural city state ally.
Standard: cC > 11, Large: cC > 20, Huge: cC > 28
3. Looking at France, with a monument cT = 4 (true also for Egypt with Monument and Burial Tomb) and with both monument and temple, cT = 7. The corresponding cC values are 9 and 16. For 9 the same as above is true, but for the case with temples, you will actually gain an increase in policy speed if you don't have at least two city state allies or a city state and a few wonders.
Standard: cC > 9 or 16, Large: cC > 16 or 28, Huge: cC > 28 or 40
4. The Songhai have the excellent Mud Pyramid, so they share the cT = 7 case with France. The same goes for adding two artists in each city.

To sum up, expanding will in almost all cases slow down your social progress. The only cases where it will speed it up are if you either aren't interested in city states and wonders, or if you play a civ with a culture bonus. The only somewhat realistic scenario where expansion could speed up your policy gain is in my opinion if you play Songhai because you'll really want the Mud Pyramid and a Monument isn't that expensive.

I'm not sure if city state bonuses scale with the map size but if they don't, expansion increasing your policy speed is a lot more likely on larger maps, probably happening at some time if you just have a monument and a temple or are playing France. If you play Songhai, or France with temples it will even happen pretty often in fact. This is another case where the game doesn't scale well (well in the sense of preserving the same effects on gameplay as on standard size) with map size.

## FoodEdit

### Food costEdit

The food cost for a city to grow is calculated as follows

$n$: The number of citizens in the city

$f$: Amount of food to grow to size n+1

$f(n) = 15 + 6 (n - 1) + (n - 1)^{1.8}$ rounded down to the next integer.

Here are some tabularized and plotted values:

Food required to grow to level n+1

Code:

 n f 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 15 22 30 40 51 63 76 90 105 121 138 155 174 194 214 235 258 280 304 329 354 380 407 435 464 493 523 554 585 617 650 684 719 754 790 826 863 901 940 979

Comment: These results do not match f(n) equation given.

Integrated food values (total amount food it takes to grow to size n)

Code:

 n f 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 15 37 67 107 158 221 297 387 492 613 751 906 1080 1274 1488 1723 1981 2261 2565 2894 3248 3628 4035 4470 4934 5427 5950 6504 7089 7706 8356 9040 9759 10513 11303 12129 12992 13893 14833 15812

What this means is that growing from size 5 to 6 costs 51 food, while growing from size 10 to 11 already costs 122 food. I would like to present a few ways of looking at this problem from different angles in the following.

### Constant Food SurplusEdit

This simplification assumes that each new citizen will work a tile that's worth 2 food and you therefore have a constant amount of food surplus that is put into growth. For a normal city without any bonuses, the amount of turns you need to grow is simply given by

$t(n) = g(n)/f$

where n is the number of citizens, f the food surplus and t the number of turns until growth, assuming you start at 0 food. Let's look at the numbers for df = 8

Code:

 n t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 2 3 4 5 6 8 10 11 13 15 17 19 22 24 27 29 32 35 38 41 44 48 51 54 58 62 65 69 73 77 81 86 90 94 99 103 108 113 118 122

This kind of explains why growth feels so slow once you hit the teens: The number of turns cities need to grow becomes pretty large, even with a pretty decent food surplus. A city takes about the same time to grow from size 12 to 13 as from size 1 to 5.

### Constant food per citizenEdit

Here, the assumption is that each citizen will yield a certain mean amount of food. To understand what happens there better, let's define another function, the average growth cost per citizen

$ga(n) = g(n)/n = 9/n + 6 + (n - 1)^{1.8}/n$

(In blue the exact function, in red the rounded one the game uses)

The last term is approximately n^0.8 and dominates for large n. As you can see, this function has a minimum at size 4 (about 3.75 in the continuous case) and then constantly grows a little less than linearly - but linear growth is an excellent approximation.

As you can see below, the function 0.41 n + 8.18 fits the data over this range perfectly but is much simpler (the functions agree at all integer places).

So how do you interpret this function? What it tells you is that, with a constant amount of food produced per citizen, your city will grow more quickly than before until it hits size 4, and then starts slowing down again. Or, to put it more simple, this function tells you the amount of food each citizen has to produce to let the city grow.

An alternative, but equivalent, point of view is that, if you're aiming for a constant city growth speed, each citizen has to become more efficient as your city grows, in an approximately linear function.

You should note that the amount of food generated per citizen will usually be larger for small cities due to the city square itself (especially with maritime CS).

### Slightly more realisticEdit

Cities start with a center square that yields some food (this catches granary, maritime city-states and water mill food, too). So the assumption that the amount of food per citizen will stay the same isn't really completely applicable. So let's introduce a new variable, f_csq, the city square food amount.

The assumption of an average amount of food per citizen now makes sense if you leave the city square food out. So let's call the amount of food each citizen generates f_c. The total food per turn, f, is then given by

$F = f_c * n + f_{csq}$

More interesting for us is the food surplus. This is given by

$f = F - 2*n = (f_c - 2)*n + f_{csq}$

The amount of surplus food produced per citizen is then

$fa = f/n = f_{csq}/n + f_c - 2$

To get the number of turns we need for growth, we need to combine this with the food growth cost and get

$t = g/f$

Since this function doesn't read too pretty, I'll just plop in some numbers and give you a graphical representation. Let's say f_csq = 2, which is the case if you don't have city state allies. If the city is to grow in a reasonable amount of time, f_c should be greater than 2.5, as you can see below

You will notice a local minimum evolving in the f_c = 4 case, which signifies the onset of the constant food per citizen approximation, which is the limiting case for f_csq = 0

### Are granaries worth it?Edit

Ultimately, that's for you to decide. I can give you some information so you can make an educated decision, though. Granaries increase f_csq from 2 to 4, so let's look at what happens to t if we make that change. Dashed are the values without granary, the full lines are with a granary

More useful to judge the granary's effects is looking at the difference between the case without a granary, and the case with a granary

As you can see, the difference quickly becomes essentially constant at a city size of 10 or more for any case of f_c > 2. From this analysis, I'd say that for f_c values up to 3, a granary is generally worth it, because it saves you a turn or two per growth step. For f_c = 2.5 it's very much worth it, and this seems to be a more realistic case than the higher values because not every citizen will work a farm (the f_c = 4 case represents each citizen working a Civil Service/Fertilizer farm) and some won't produce any food at all, like specialists.

### Other f_csq effectsEdit

We can continue this analysis by increasing f_csq in steps. For example, a maritime CS will yield 2 extra food, as will a water wheel. Let's see what happens if we go from f_csq = 4 to f_csq = 6. I will omit the more unrealistically high cases from now on for a better overview. Shown is again the total number of turns needed and the difference between 4 and 6. This time, f_csq = 4 is dashed

So getting the second city-state isn't as good as getting the first, and getting a granary when you already have a city-state isn't so great, either. It still shaves off a turn (or three in the f_c = 2.5 case), though.

The next increase step, from 6 to 8 (the "before" being dashed as usual)

Now things start becoming somewhat underwhelming. As I said, if things are worth it for you is up to you to decide, but I'd definitely not build that water mill if I already have a granary and a city-state ally because the difference will only be something like two turns in three growth steps or so, which isn't exactly a lot.

### f_c values < 2Edit

After reading a comment from ehrgeix, I think it makes sense to extend the analysis to f_c values that are smaller than 2. The values greater than 2 are applicable if you want to let your city continue to grow for the rest of the game. Values smaller than 2 still make sense in transitionary periods, if you want a growth cap, or if you're fine with your growth speed slowing down even more than in the constant food surplus case discussed above.

Qualitatively, we can already see from looking at $f = (f_c - 2)*n + f_{csq}$, which occurs as the denominator in the formula for t, that these functions will have a pole at a finite n, because f_c - 2 becomes negative. The locus of this singularity is given by $n = f_{csq}/(2 - f_c)$. This singularity signifies the number of citizens where the city stops growing.

In the following are some graphs in the same way as above. First, f_csq = 2

If you increase f_csq you shift the position of the pole to the right, so the difference between t(csq = 2) - t(csq = 4) has a singularity at the same points as t. See the f_csq = 4 case below. Dashed is the "previous", in this case f_csq = 2

f_csq = 4 -> f_csq = 6

f_csq = 6 -> f_csq = 8

## Unit maintenanceEdit

As of 6/2015, the formula for unit maintenance as defined in all versions CvGameCoreDLL is calculated as follows:

• b = Base unit cost (INITIAL_GOLD_PER_UNIT_TIMES_100)
• f = Free units from handicap (GoldFreeUnits), specific unit types, or policies
• u = Total paid units
• n = max(0, u-f) = Number of actual paid units (if u-f is less than 0 then n = 0)
• m = Multiplier (UNIT_MAINTENANCE_GAME_MULTIPLIER)
• d = Divisor (UNIT_MAINTENANCE_GAME_EXPONENT_DIVISOR)
• t = Current turn
• e = Estimated end turn (based on all entries for the current GameSpeed in GameSpeed_Turns)
• g = t/e = Game progress factor
final cost = (n*b(1+g*m)/100)(1+(g/d))


Unit cost modifiers from traits are applied before the exponent. Unit cost modifiers from policies, handicap, and AI difficulty level are applied to the final cost.

Since there's no easy "each unit in turn t will cost this much" here's a table you can use as a rough reference. The first row is the number of turns, the first column the number of units

Total unit maintenance costs
Turns
Units 1 20 50 100 150 200 250 300 350 400
4 22357911141720
8 45711151924303643
12 681116233038475768
16 8101422314152647894
20 1013182839526682100121
24 12162234476380100122148
28 14182640567494118145176
32 162130466485109136168204
36 182434527396124155191232
40 2026375881108138174215261
44 2229416489119153193238291
48 2432457098131168212262321
52 26354976107142184231287350
56 28375382115154199251311381
60 30405788124166214270335411
64 32436194133177229290360442
68 344564100141189245310385473
72 364868106150201260330410504
76 385172112159213276349435535
80 405476118167225292370461567
84 425680124176237307390486598
88 445984130185249323410512630
92 466288137194260339430537662
96 486492143203273354451563694
100 506795149211285370471589727

Here's an equivalent table detailing the cost per unit

Maintenance cost per unit
Turns
Units 1 20 50 100 150 200 250 300 350 400
4 0.500.500.751.31.82.32.83.54.35.0
8 0.500.630.881.41.92.43.03.84.55.4
12 0.500.670.921.31.92.53.23.94.85.7
16 0.500.630.881.41.92.63.34.04.95.9
20 0.500.650.901.42.02.63.34.15.06.1
24 0.500.670.921.42.02.63.34.25.16.2
28 0.500.640.931.42.02.63.44.25.26.3
32 0.500.660.941.42.02.73.44.35.36.4
36 0.500.670.941.42.02.73.44.35.36.4
40 0.500.650.931.52.02.73.54.45.46.5
44 0.500.660.931.52.02.73.54.45.46.6
48 0.500.670.941.52.02.73.54.45.56.7
52 0.500.670.941.52.12.73.54.45.56.7
56 0.500.660.951.52.12.83.64.55.66.8
60 0.500.670.951.52.12.83.64.55.66.9
64 0.500.670.951.52.12.83.64.55.66.9
68 0.500.660.941.52.12.83.64.65.77.0
72 0.500.670.941.52.12.83.64.65.77.0
76 0.500.670.951.52.12.83.64.65.77.0
80 0.500.680.951.52.12.83.74.65.87.1
84 0.500.670.951.52.12.83.74.65.87.1
88 0.500.670.951.52.12.83.74.75.87.2
92 0.500.670.961.52.12.83.74.75.87.2
96 0.500.670.961.52.12.83.74.75.97.2
100 0.500.670.951.52.12.93.74.75.97.3
Civilization V 
Gods & KingsBrave New World

Concepts
City-StateCultureEspionage‡ • FoodGoldGreat PeopleHappinessProductionReligion‡ • ScienceTourism

Guides etc.
AchievementsMathematicsModdingMajor PatchesSoundtrackCivilopedia

† Only in vanilla Civ5
‡ Only in Gods & Kings and Brave New World