A small summary of game Stone Games

A) Ba Shi game (Bash game): Only a bunch of n items, two people take turns from this pile of goods, the provision of at least one at a time, the maximum number of M. Finally the light winner wins.

It is easy to think of when n% (m+1) <>0, first take the win, first take n% (m+1), after each turn to keep two people take the sum of items for m+1 can.

This game can also have a disguised play: Two people turn off, at least one at a time, up to 10, who can report 100 wins.

(ii) Witzov games (Wythoff game): There are two piles of various items, two people take turns from a heap or at the same time from the two piles of the same number of items, the provision of at least one at a time, more than open, and finally the winner of the light.

If a face (0,0), then A has been lost, this situation we call the singular situation. The first few strange situations are: (0,0), (3,5), (4,7), (6,10). As you can see, A0=b0=0,ak is the smallest natural number that has not been seen before, and Bk=ak+k.

So let's give a situation (a, B), how do you judge if it is a singular situation? We have the following formula:

AK =[k (1+√5)/2],bk= AK + k (k=0,1,2,...,n square brackets denote rounding function)

The wonderful thing is that the number of golden Divisions (1+√5)/2 = 1 appears. 618 ..., therefore, the rectangle composed of AK,BK is approximately the golden rectangle, because 2/(1+√5) = (√5-1)/2, you can first find J=[a (√5-1)/2], if A=[J (1+√5)/2], then a = AJ,BJ = AJ + j, if not equal, then A = aj+ 1,bj+1 = aj+1+ J + 1, if not, then it is not a singular situation. Then according to the above-mentioned law, we will encounter strange situation.

(iii) NIM game (Nimm game): There are three piles of various items, two people take turns from a heap of any number of items, the provisions of at least one at a time, more than a few, the last person to win the light.

For any singular situation (a,b,c), there are a^b^c=0.

Non-singular situations (a,b,c) (a<b<c) are converted to singular situations by simply changing C to a^b, i.e. subtracting C-(a^b).

A small summary of game Stone Games