If opponent only can move between the two diagonal squares in the "open" corner, does he lose? How many times can he makes these fruitless moves?
123073
What is the term for a stalemate in Checkers?  What are the official conditions for a "no winner game"?
119871
Moving Around Other creatures      You can move through a nonhostile creature's space [but] you can't willingly end your move in its space.   Is that "end your move" at the end of your turn or if you stop to attack? I had a player who wanted to move into another friendly character's space, attack a hostile, then sidestep behind a wall.
8926
I came across a question yesterday about combinations, and I wanted to know what the correct answer was. The question states as follows:  There are 8 spaces that are alternately black and white. There is one king, one queen, two identical rooks, two identical bishops, and two identical knights. The king needs to be surrounded by the two rooks. Then, the two bishops can be put on any of the remaining spaces as long as one is on a black space and the other is on a white space. Finally, the queen and the knights can be placed anywhere in the remaining spaces in relation to the other pieces. How many possible arrangements are there considering these conditions?
73673
You are given four points (on a Euclidian plane) that make up the corners of a square. You may change the positions of the points by a sequence of moves. Each move changes the position of one point, say p, to a new location, say p', by "skipping over" one of the other 3 points. More precisely, p skips over a point q if it is moved to the diametrically opposed side of q. In other words, a move from p to p' is allowed if there exists a point q such that $q = (p + p') / 2$.  Find a sequence of moves that results in a larger square. Or, if no such sequence is possible, give a proof of why it isn't possible. (The new square need not be oriented the same way as the original square. For example, the larger square may be turned 45 degrees from the original, and the larger square may have the points in a different order from in the original square.)  I've tried many different combinations and still haven't gotten anything more than 3 points in the right position and one point completely off. Is this possible?
121401
Two rooks are placed on an 8 by 8 chessboard such that they're in different squares. What is the probability they can't attack each other? Rooks can attack each other if and only if they're in the same row or column. I know that there are questions like this but they give the general numbers not the probability I am not sure how to start the questions and I have looked at this questions for hours. Any help would be welcome.
39348
Can you resolve a dispute between myself and my girlfriend? She recently won a game with what I think is a double movement (although there's no mention of this in the rule book).  She put one two letters down to make one word, which also scored linked to another letter. But she then added another letter immediately to the left of the finished word, which then extended the second word. I just don't think you can make a second movement like this in scrabble!    My sister placed down W, D and E, the ones that I circled in red. Is she allowed to do that? I told her it was impossible is it?  Can you resolve this for me please?
51714
We have a $3\times3\times 3$ cube which has $27$ cubes each $1\times1\times1$ stuck together as usual. $2$ cubes are neighbours if they have a common face. The corner cubes are the $8$ cubes at the corners of the big cube. Can the worm start from a corner cube and eat its neighbour cube, continuing until at the end the last cube eaten is the one in the center of the big cube which has 6 neighbour cubes?
33117
Your (honest) opponent choose a random number from 1 to 13 inclusive. You have to guess the number, and you win if the guess is correct. If not, your opponent either reduces the number chosen by one or increases it by 1, and you guess again.  The question is, what is the minimum # of attempts necessary to guarantee a win for you.  I am not able to get a handle on the problem.  Also, (a new variant just thought of), how many guesses should be allowed for a fair or "nearest to fair" game ?
112366
Consider an $n\times n$ chessboard whose top-left corner is colored white. But Alice likes darkness, so she wants you to cover those white cells for her. The only tool you have are black L-shaped tiles each of which covers $3$ unit cells. Formally, each tile covers unit cells satisfying the following:  Two of the cells are adjacent to the third (shares a side). All three of the cells do not lie on the same row or same column. No two tiles should overlap (cover the same cell) or go outside the board.  Since these tiles cost a lot, you have to cover all the white cells using the minimum number tiles. Example: $1\times 1$ Answer: Impossible, there's a single cell which is white. Since one tile needs $3$ empty cells, there's no way to cover this cell. Example: $4\times 4$ Answer: $4$ ($4$ tiles can be placed as shown)  Example: $7 \times 7$ If each tile can be represented by a number, and each uncovered piece of board can be represented by 'zero', then the answer for a $7 \times 7$ board is $16$: $$ \begin{bmatrix} 16&amp; 16&amp; 15&amp; 15&amp; 14&amp; 14&amp; 13 \\ 16&amp; 12&amp; 15&amp; 11&amp; 14&amp; 13&amp; 13 \\ 12&amp; 12&amp; 11&amp; 11&amp; 10&amp; 10&amp; 9 \\ 8&amp; 8&amp; 7&amp; 6&amp; 10&amp; 9&amp; 9 \\ 8&amp; 7&amp; 7&amp; 6&amp; 6&amp; 2&amp; 2 \\ 5&amp; 5&amp; 4&amp; 3&amp; 3&amp; 1&amp; 2 \\ 5&amp; 0&amp; 4&amp; 4&amp; 3&amp; 1&amp; 1\\ \end{bmatrix} $$ Question For any given $n$, what will be the minimum number of tiles? (Note: Answer exists for odd value of $n \geq 7$)
64633
Two players, Alex and Brad, take turns removing marbles from a jar which initially contains $2015$ marbles. Assume that on each turn the number of marbles withdrawn is a power of two.  If Alex has the first turn and the player who takes the last marble wins, is there a winning strategy for either of the players?    I try to solve it like the follow:  First) I assume that both player know the amount of marbles they took each other  Second) The problem say they pick power of 2 then the possible picks are $[1 , 2 , 4, 8, 16 ,32,64 128 , 256 , 512 , 1024]$ there are only $2015$ marbles.  Third) the game has 2 players.  Continue:  a)  The only odd pick is $1$ due to $2^0$ is $1$. The rest are odds  If player A pick odd (1) B must pick an even number and A continue pick even and also B until complete the $2015$.  And  If player A pick even (2, 4, etc) B must pick an odd (1) number and A continue pick even and also B until complete the $2015$.  I cannot see how can I use 2015 to set a winning rule, I also try use binary but I couldn’t do any advance.  Please, let me know what I am doing wrong and how I can continue. Thanks, Greg Martin and Shailesh for your help.  Note: I try to use brute fore (excel ;-) ) but I see that multiple of 3 are the wining but I can’t not put my assumption in words with the $2015$ number.  Thanks again.
21054
I'm having trouble fully understanding this introductory combinatorics problem; this is all that's provided, so it's most likely me not fully comprehending the question.  Any help is appreciated.  The problem is this:     Each square of a 1998 by 2002 chess board contains either 0 or 1 such that the total number of squares containing 1 is odd in each row and each column. Prove that the number of white unit squares containing 1 is even.   I have discovered that this is a legitimate question and has been copied out of a book entitled 102 Combinatorial Problems where the problem and solution are clearly stated . Thank you and apologies for the ambiguity.
81620
In a game, you are standing somewhere and a move consists of either moving forward any number of steps, not moving at all, or moving $1$ step backwards. If you complete $n$ moves, find the number of sequences that end with you in your starting position and which never go behind your starting position. If the &quot;never go behind the starting position&quot; condition weren't there, the answer would just be $\binom{2n-1}{n-1}$. However I'm not on sure how to proceed from here.
101411
There is an unfair coin. It tends to land on one side more than on the other. It is unknown which side is it.  There is Mr. A and Mr. B. They argue about something and they want to use that coin to decide who is right.  Is there any technique of using that unfair coin to get fair result?  My solution: I can think of only that there is going to be 2 rounds. First time Mr. A chooses HEAD, than Mr. B chooses HEAD if any of them wins twice they win.
51503
In Sorry, if you are 7 spaces away from your Home and you draw a 12 (and your other pawns are in Start or Home), do you have to move past your safety zone like you're going around the board a second time or do you forfeit your turn?
61537
OEIS sequence  (based on ) describes the     minimum number of dominoes on an $n \times n$ chessboard to prevent placement of another domino.   The sequence begins:  0, 2, 3, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75   Fifteen terms are known, and a few folks have conjectured that $$    A280984(n) = \left\lceil \frac {n^2}3 \right\rceil \text{ for } n &gt; 1. $$  Andrey Zabolotskiy has a proof that (1) this is a lower bound, and (2) this conjecture holds when $n$ is a multiple of $3$.  In order to prove the conjecture it is sufficient to give a tiling strategy for the $(3n+1)\times(3n + 1)$ and $(3n+2)\times(3n+2)$ boards.  Example of $3n \times 3n$ boards.  To achieve this bound for the $3n \times 3n$ boards you can just do the naive placement, but for the other boards, it's not clear which placement strategy to use.  (Notice that the second and third examples are not optimal: the last two boards have $35$ tiles and $42$ tiles respectively, but there exist $34$ and $41$ tile placements.)      Question  Is there simple strategy for the $(3n+1) \times (3n+1)$ and/or the $(3n+2) \times (3n+2)$ boards that attains the lower bound? Or is the conjecture false?
103130
Perhaps, this game is already known, but I did not find anything about it, I call  it "knight".  The rules :   Player 1 chooses the starting square of a knight on a normal 8x8 - chessboard.  The players alternately move the knight to a square which was not already   visited. The player having no more move loses.  Which player has a winning strategy and why ?
45787
How useful is that rule and how many times we can use that in one game?  Under what conditions can the en passant rule be used?  How often does it come up and is it a good idea to use it?