Recently, I have done a lot of problems in mathematics , It's settled The basis of number theory , linear algebra （ Only the matrix , The determinant holds the pit ）, Combinatorial mathematics Some simple questions in . So the inevitable next step is to Game theory this philosophy Work starts in Tai Keng .

Of course , Because I am very happy food , So we can only start from the most basic and easiest part .

This time, we mainly introduce two basic models ：Bash Game&&Nim Game, It's more classic

### 1.Bash Game

Bash Game( Bash Game ) It's a kind of Zero basis Introduction to the topic , Its basic model is like this ：

A pile of stones n individual , They take turns taking stones from it , It is stipulated that at least one , Most take m individual , The last winner wins . Ask two people about the game , They all use the smartest strategies , Who will win in the end .

First of all, let's start from childhood ：

• When m>=n when , You can catch the stones first , such The first step is to win

• When n=m+1 when , First hand, no matter how you catch it , Will stay 1~m A stone , In this way, you can finish it with one hand , such If you start, you'll lose

• When m+1<n<2*(m+1) when , The first thing you need to do is grab the right stones and you'll have the rest m+1 A stone , And then you lose , that The first step is to win

• When n=2*(m+1） when , The backhand can take this strategy , When you start x Time , Catch it in the back m+1-x individual , This will put m+1 I'd like to give you the first hand , that If you start, you'll lose

• ......

So from the above analysis, we can easily conclude that ： When n%(m+1)==0 when , If you start, you'll lose ; Otherwise, the first player will win

This is very simple , Because who will face it first k*(m+1)(k As a positive integer ) This situation is bound to fail . Because one side can let you two get stones and for each time m+1 Cause the other side to fail .

There are still some distortions to this problem , The classic is ： Take a number from 0 It's starting to grow , The scope of each increase is 1~k, Who first increases the number to n Just Win||Lose

This translates directly into Bash Game Just do it

Here are some examples &&CODE（ The following examples are from HDU

1846

Bash The board problem of game , Just go straight on

CODE

``````#include<cstdio>
using namespace std;
int t,n,m;
inline char tc(void)
{
static char fl[100000],*A=fl,*B=fl;
}
{
x=0; char ch=tc();
while (ch<'0'||ch>'9') ch=tc();
while (ch>='0'&&ch<='9') x=x*10+ch-'0',ch=tc();
}
int main()
{
//freopen("CODE.in","r",stdin); freopen("CODE.out","w",stdout);
while (t--)
{
puts(n%(m+1)?"first":"second");
}
return 0;
}
``````

2149

This is the case with the variant , We're doing a transformation. We're doing a transformation , As for the output scheme, we can make a special judgment

CODE

``````#include<cstdio>
using namespace std;
int n,m;
inline void write(int x)
{
if (x/10) write(x/10);
putchar(x%10+'0');
}
int main()
{
register int i;
while (scanf("%d%d",&m,&n)!=EOF)
{
if (n>=m)
{
for (i=m;i<n;++i)
write(i),putchar(' '); write(n); putchar('\n');
} else
if (m%(n+1)) write(m%(n+1)),putchar('\n'); else puts("none");
}
return 0;
}
``````

2138

ditto ,Bash Game Just go straight on

CODE

``````#include<cstdio>
using namespace std;
int t,n,m;
inline char tc(void)
{
static char fl[100000],*A=fl,*B=fl;
}
{
x=0; char ch=tc();
while (ch<'0'||ch>'9') ch=tc();
while (ch>='0'&&ch<='9') x=x*10+ch-'0',ch=tc();
}
int main()
{
//freopen("CODE.in","r",stdin); freopen("CODE.out","w",stdout);
while (t--)
{
puts(n%(m+1)?"Grass":"Rabbit");
}
return 0;
}
``````

4764

Pay attention to the greater than or equal to n You lose. , therefore n One less

CODE

``````#include<cstdio>
using namespace std;
int n,m;
inline char tc(void)
{
static char fl[100000],*A=fl,*B=fl;
}
{
x=0; char ch=tc();
while (ch<'0'||ch>'9') ch=tc();
while (ch>='0'&&ch<='9') x=x*10+ch-'0',ch=tc();
}
int main()
{
//freopen("CODE.in","r",stdin); freopen("CODE.out","w",stdout);
for (;;)
{
if (!n&&!m) break;
puts((n-1)%(m+1)?"Tang":"Jiang");
}
return 0;
}
``````

### 2.Nim Game

This is actually Bash Game The evolution of , But the depth of thinking has gone up several steps

First, let's look at the basic model ：

There are... On the ground n Rubble , Each person can take out any number of stones from any pile of stones at a time and throw them away , You can finish it , Can't do without taking . You can only take it from a pile at a time . In the end, those who don't have a stone will lose . If a is the first , And tell you this n The number of stones , He wants to know if there is a winning strategy .

Isn't that complicated , First of all, let's consider the case of small data ：

• When there's only a pile of stones , Unless there are no stones , Otherwise, we can grasp it first

• When there are two piles of stones , The backhand only needs to follow the action of the forehand

• ......（ I can't figure out what's going on ）

But before us, there was a great God who successfully proved that The magic conclusion

When n The number of heaps of stones is equal to 0 when , If you start, you'll lose , Otherwise, the first player will win

Please move to attack dalao's blog

I'm confused, right? In fact, I'm also , So let's Perceptual understanding once （ The following description is for personal understanding only , Hardly any Any theoretical basis , Please choose to read ）

First of all, because it's all 0 I failed when I was young , So we can think that when their XOR sum is 0 Namely Be defeated

In this way, we can start with if XOR and for 0 Then you lose

If not , We can win in a dirty way （ That's the way it's mentioned above ）

We can visualize it as And Bash Game First in the middle, first in the middle k*(m+1) And then by making and always for m+1 To win

And that's the magic Nim Game

Here's a template question Luogu P2197 nim game

Because it's naked Nim Game, Right here CODE 了

``````#include<cstdio>
using namespace std;
int t,n,x,tot;
inline char tc(void)
{
static char fl[100000],*A=fl,*B=fl;
}
{
x=0; char ch=tc();
while (ch<'0'||ch>'9') ch=tc();
while (ch>='0'&&ch<='9') x=x*10+ch-'0',ch=tc();
}
int main()
{
//freopen("CODE.in","r",stdin); freopen("CODE.out","w",stdout);
while (t--)
{
puts(tot?"Yes":"No");
}
return 0;
}
``````

Of course, this is just the beginning of game theory , Something really powerful, like SG function &&MINMAX I'm still confused when I wait for something

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