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Introduction
The focus of this topic is the card game called Three Kingdoms Kill (3KK for in short), which is a popular Chinese card game based on the Three Kingdoms period of China and the semi-fictional novel Romance of the Three Kingdoms (ROTK for in short). The rules of 3KK originally come from the Italian card game Bang!
I am going to talk about the relationship between this game, 3KK, and mathematics. Whether people can use the science of probability and statistics to win the game?

The benefit of each card
In
3KK, every card has its special use. Some of them can steal the hand from another
character, while some can recover health points. The value of each card is also
different. Therefore, we can classify the cards into “high value” and “low
value”.
If I measure the value
of one card as 1 unit and the value of one health point as 2 units, I can
measure the value of all the cards in the deck by number. (The rule “Health
Point ratio to each card = 2 units to 1 unit” comes from Kayak, the original
game designer)
I list the value of
most of the cards and show it below:
Attack shift between 1 and 2
Dodge 2+
Peach 2
Wine shift
between 0 and 2
Dismantle 1
Steal 2
Duel shift between -1 and 2
Duress shift between 2 and 3
Draw2 2
Negate 1
Blaze shift between 0 and 1
Ration Depleted
1.5
In the list above, the
cost of the card itself should also be subtracted. Therefore, I will subtract 1
unit for each card. After that, I define the average value of each card that is
equal to one unit or more than one unit as “high value”.
The “dodge” card is an
exception. Although the value of “dodge” is always higher than 1, because the
card “dodge” cannot be played proactively, its real value is lower. Therefore,
it is a low value card.
The
value of some other cards such as equipments is hard to calculate. The value of
those cards depends on the situation. For example the armor cards have no value,
but they have the ability to defend, they could be very important. Therefore,
they can be high value cards. Acedias and +1horses cards could also vary in value.
As a result, the entire deck of the 160 cards is as follows:
High
value cards: 12+5+2+4+3+6+4=36
The
percentage of high value cards in a deck is: 36/160 = 0.225 = 22.5%
An
equation shows the odds:
1/X
= 22.5/100
X
= 4.45
This
data means the player could get one high value card from each 4.45 drawing;
which also means if a character can additionally draw 4.45 cards, he/she can on
average get one high value card. I will talk about that in the latter paragraph.
The prediction of
drawing
According
to the rules of the game, each character can draw two cards from the deck each
round. Nevertheless, because of the high value cards, the value will be higher
than 2. Therefore, I will calculate the average of each draw.
To
simplify the procedure, I assume the deck is infinite and define the benefit of
all the low value cards as 1 unit and the benefit of all the high value cards
as 2 units. According to the calculation above, the chance of drawing a high
value card is 22.5%. Also, there are two cards drawn in a round. Therefore, I
get the relationship below:
total income
first card
second card
percentage
2 units
1 units
1 units
p1*p1
0.600625
3 units
1 units
2 units
p1*p2
0.174375
3 units
2 units
1 units
p2*p1
0.174375
4 units
2 units
2 units
p2*p2
0.050625
p1 =
percentage to get the low value card = 77.5%
p2 =
percentage to get the high value card = 22.5%
The
table shows the percentage. To get 2 units is 60%, to get 3 units is 35% and to
get 4 units is 5%. I use the equation below to get the average:
2*(p1*p1) + 3*(p1*p2)*2 + 4*(p2*p2)
=2.45 (units)
Hence,
the average value from drawing is 2.45 units/ round. However, the practical
condition is more uncontrollable. The range shall shift depending on the
characters and the use of cards.

The features of the
characters
The different
characters are the most attractive part of 3KK. Everyone who knows this game
must have his or her own favorite characters. However, some of the characters
have more supporters and some have less; one important reason is because
different characters also have different strengths. I can calculate the benefit
of the characters per round so that I can measure them in the “scientific” way.
For example, there are the abilities of the character called Zhen Ji
who has 3 health points.
Character
ability 1: Impetus for War 倾城Every hand that is a "club" or a "spade"
suit can be used as a dodge.
Character ability 2: Goddess Luo 洛神At the beginning of your turn, you can flip a judgment card.
If the judgment card is a black-suited card ("clubs" or
"spades"), you can keep the card in your hand and can continue to
flip over another judgment card. Once the judgment card is a red-suited card,
the ability ends and you cannot keep that red-suited card.
The first ability of Zhen Ji is to
transform the cards to a dodge.
According to the calculation in the last paragraph, we already know that the value
of using a dodge is more than 1
unit. That means this ability can transform a lower value card to a higher
value card. The value of this ability depends on whether the other characters
use attack on you or not. According
to the percentage of attack cards in
the deck, I assume that Zhen Ji will be attacked once per round. Therefore,
there will be one lower value card transformed to a higher value card. The
value-difference is 2 - 1 = 1 unit. However, as I said before, because the dodge cannot be played proactively, the
value of this ability is lower than what the data shows.
The
second ability is a drawing ability. From observing, I can tell that the value equation
of this ability is a geometric progression. Therefore, I can use the geometric
sequence summation equation:
S = a1 * q^ (n - 1) / 1 – q
Set
a1 = 1;
q = ½;
n = infinite
S = 1 * ½^ (n – 1) / 1 - ½
S = 2 * ½^ (n – 1)
∵ lim n→∞ [½^(n – 1)] = ∵ ½ * lim n→∞ (½ ^n)
∴ S = 2 * ½ = 1
Then, I get the average value of the
second ability is 1 units/ round.
Consequently,
the benefit of Zhen Ji is available now. If I assume the value of ability
“impetus of war” is 0.5 units, the average of Zhen Ji’s benefit is 1.5 units/
round.
For
comparison, there is another character named Sun Quan, who has 4 health points.
His abilities are shown below.
Character
ability 1: "Balance of Power 制衡" During
your action phase, you can discard any number of cards and draw the same number
of new cards from the deck. This skill is limited to once per round.
Character
ability 2: [Ruler ability, Enforced ability] "Rescued 救援" When you are rescued from the brink of
death by other Wu characters via the use of peach, you recover 1 additional unit of
health (from zero health to two units of health).
Because the second ability
is not used often, I will simplify by skipping it. Seemingly, the value of the
first ability is easy to count. The cost = the benefit, the value of this ability
equals zero. However, this character is the strongest at the “1 versus 1” mode.
The reason is the existence of high value cards. Sun Quan
can transform the low value or non-value (because it cannot be use immediately)
cards to random cards. According to the calculation I did above, once Sun Quan
draw 4.45 cards, he can get 1 unit of benefit. I assume that the value of every
non-value card is 0.5 units. Sun Quan on average transforms 1 non-value card (0.5
units) and 1 low value card (1 unit) to 2 random cards (2.45 units) per round.
The cost is about 0.5 + 1 = 1.5 unit, therefore:
2.45–1.5
= 0.95 units/ round
The average benefit of Sun Quan is around 0.95
units per round.
The character Zhen Ji
has higher benefit than Sun Quan, so the way to balance it is by using via
different health points. The characters that have the weaker ability usually
have higher health points.
I can use the same method
as the examples above to get the average benefit of all the characters. However,
that will be a lot work, so I searched for the information on the internet. Someone
else has figured that out. He stated that the average benefit of characters
with 3 health points is around 1.5 united, and the benefit of characters with 4
health points is around 0.7 units.
As you see, the
character Zhen Ji has the benefit equal to the average. Thus the strength is in
the normal level. Sun Quan has the benefit higher than the average. Therefore he
is what we call a “strong character”.

求学术帝发中文版

英语不好的给跪

介绍
这个话题的焦点是卡片游戏叫三国杀(3
kk在短),这是一个受欢迎的中国纸牌游戏基于中国三国时期和半虚构的小说三国演义(ROTK在短)。规则3 kk最初来自意大利的卡片游戏爆炸!
我要谈论这个游戏之间的关系,3 kk,和数学。人们是否可以使用科学的概率和统计来赢得这场比赛吗?

每张卡片的好处
在
3 kk,每张卡都有其特殊的使用。他们中的一些人可以偷的手从另一个
字符,而一些可以恢复健康点。每张卡片的价值也
不同的。因此,我们可以把卡片进入“高价值”和“低
价值”。
如果我测量值
一个卡1单元和一个健康的价值点为2单位,我可以
测量值在所有的卡片在甲板的号码。(规则”的健康
点比每个卡= 2单元1单元”,原来自Kayak
游戏设计师)
我列出的值
大部分的卡片和显示如下:
1和2之间的攻击转变
道奇2 +
桃2
酒转变
介于0和2
拆除1
偷2
1和2之间的决斗转变
2和3之间胁迫转变
Draw2 2
否定1
火焰移在0和1之间
配给耗尽
1.5
在上述列表中,
卡本身的成本还应减去。因此,我将减去1
单位为每个卡片。在那之后,我定义的平均价值每个卡片
等于一个单位或超过一个单位为“高价值”。
“道奇”牌是一个
例外。虽然“道奇”的价值总是高于1,因为
卡“道奇”不能扮演主动,它的真正价值较低。因此,
这是一个低价值的卡片。
这个
其他一些卡片的价值如设备难以计算。的价值
那些卡片取决于实际情况。例如甲卡没有价值,
但是他们有能力保卫,他们可能是非常重要的。因此,
他们可以是高价值的卡片。Acedias和+ 1马牌也可能不同值。
因此,整个甲板的160卡如下:
高
值卡:12 + 5 + 2 + 4 + 3 + 6 + 4 = 36
这个
比例的高价值卡在甲板是:36/160 = 0.225 = 22.5%
一个
方程显示的可能性:
1 / X
= 22.5/100
x
= 4.45
这
数据意味着玩家可以得到一个高价值的卡片从每个4.45图;
这也意味着如果一个角色可以另外画4.45卡,他/她可以吗
得到一个高的平均价值卡。我将讨论,在后面的段落。
预测
图纸
根据
游戏规则,每个字符可以画两张牌从甲板上每个
圆的。然而,由于高价值卡,价值会更高
比2。因此,我将计算平均每一画。
到
简化程序,我假设甲板是无限的和定义的好处
所有的低价值的卡片作为1单元和造福所有的高价值的卡片
2单位。根据上面的计算,图纸的机会高
值卡是22.5%。同样,有两个牌在一个圆。因此,我
得到下面的关系:
总收入
第一卡
第二张牌
百分比
2单位
1单位
1单位
p1 * p1
0.600625
3单位
1单位
2单位
p1 * p2
0.174375
3单位
2单位
1单位
p2 * p1
0.174375
4单元
2单位
2单位
p2 * p2
0.050625
p1 =
百分比获得低价值卡= 77.5%
p2 =
百分比来获得高价值卡= 22.5%
这个
表显示的百分比。得到2单位是60%,获得3单位是35%和
得到4单位是5%。我使用下面的方程得到平均:
2 *(p1 * p1)+ 3 *(p1 * p2)* 2 + 4 *(p2 * p2)
= 2.45(单位)
因此,
从图纸的平均值是2.45单位/圆。然而,实际
条件是无法控制的,更范围应当转变取决于
角色和使用信用卡。

英文差的跪了……

大王明明说给我看的

Ctrl+F4

让吾等英语初中三级都没过的战五渣情何以堪……

大概能看懂，但是懒得看了，大学毕业后再也不想碰英语的路过