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# Playing Chess with a Coin

## Question

Will it be more interesting to play none turn-base chess with balanced skill and luck? With all regular rules applied, starting with initial position, the first move depends on head or tail result of the coin toss. Next move likewise and so on until white/black wins or draw. While in case a player's King is in check, it is his turn to make a move unless it is a checkmate.

Suppose I want to help win against myself in five or less coin spins, what is the best probability that I can make a checkmate?

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Spoiler

Probability = 1/(2*20*2*2*20*2) = 1/6400 = 0.00015625 at best along the following line ...

1. Player A (say White) wins first toss (P = 1/2)

2. W moves pawn to f4 (P = 1/20 possible starting moves for W using a pawn or a knight)

3. Player B (say Black) wins 2nd toss (P = 1/2).

4. B moves pawn to e6.

5. W wins 3rd toss (P = 1/2).

6. W moves pawn to g4 (P = 1/20 possible moves = 19 for a pawn or a knight + 1 for K to f2).

7. B wins 4th toss (P = 1/2).

8. B moves Queen to h4.

[Black checkmates on his 2nd move after 4 tosses]

Edited by rocdocmac
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I’d say you have at least a 68.75% (11/16) chance of winning against yourself. Because you can make “stupid” moves when helping yourself win, right?

So black can win against white in three moves if they go like this:

1. W: Pawn to F4 (or F3)

2. B: Pawn to E5 (or E6)

3. B: Queen to H4

White needs to win the first coin toss (probability 1/2) and black needs at least two of the remaining four (probability (1/2)^5 * 11 = 11/32 ).

The same goes for black winning against white, so the first toss decides which side will try to win.

The probability then would be 2 * 11/32 = 11/16 = 68.75%. At least that much, because I haven’t explored all possibilities.

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Spoiler

s

Since the initial position is symmetrical the first winner from coin toss does not matter. What are the moves lines  so that the "first to move" side should have the most possible checkmate on or before its 5th move.

Edited by TimeSpaceLightForce
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Welcome here @ Silver.

Note that the fool's mate requires 4 tosses

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Ah-ha! The first toss is irrelevant. Thus the second player has the best chance ...

Spoiler

My first attempt can then be bettered to 1/3200 rather than 1/6400?

Maximum probability for 2nd player to checkmate would then be then 0.0003125 (~0.03%) on his second move.

But here only for a "2nd to move" win.

If it's a matter of "first to move" to win, then it may look different.

NO - I'm still missing something here ... will start all over!!!!!

Edited by rocdocmac

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