tag:blogger.com,1999:blog-22372446372288095912014-10-06T20:15:22.169-06:00Games ConceivedSimple games invented by Leroy Quet.Amorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.comBlogger149125tag:blogger.com,1999:blog-2237244637228809591.post-70796734119210854152011-08-17T13:13:00.001-06:002011-08-17T13:13:52.277-06:00Sum Of Products, Number Of Numbers GameHere is a game for 2 players played using an n-by-n grid drawn on paper.
<br />
<br />Players take turns. On a turn a player writes any one integer, 1 to n, into any empty square of the grid.
<br />
<br />After n^2 total turns, the game is over.
<br />
<br />Player 1 gets the sum of the scores for each row. The score for a row is (the number of 1's in the row) * (the number of 2's in the row) * (the number of 3's in the row) *...(the number of m's in the row), where m is the largest integer such that all integers 1 through m occur in that row.
<br />
<br />Player 2 gets the sum of the scores for each column. The score for a column is (the number of 1's in the column) * (the number of 2's in the column) * (the number of 3's in the column) *...(the number of m's in the column), where m is the largest integer such that all integers 1 through m occur in that column.
<br />
<br />A player gets 1 point for a row/column if there are no 1's in that row/column.
<br />
<br />The player with the largest score wins.
<br />
<br />And example game:
<br />
<br />1 1 2 1 2 5
<br />1 3 2 1 3 5
<br />5 1 1 1 3 1
<br />1 2 1 2 1 2
<br />2 1 1 2 1 1
<br />2 3 5 3 4 1
<br />
<br />Player 1 gets:
<br />3*2 + 2*1*2 + 4 + 3*3 + 4*2 + 1*1*2*1*1
<br />=
<br />33 points.
<br />
<br />Player 2 gets:
<br />3*2 + 3*1*2 + 3*2 + 3*2*1 + 2*1*2*1 + 3*1
<br />=
<br />31 points.
<br />
<br />Player 1 wins.
<br />
<br />Thanks,
<br />Leroy Quet
<br />Amorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-47005902815484359772011-08-17T13:11:00.000-06:002011-08-17T13:12:27.052-06:00Binary Scramble GameHere is a game for any plural number of players.
<br />
<br />A list of 0's and 1's is written on a piece of paper by the players taking turns, each player appending a 0 or a 1 onto the right side of the list each turn. After a predetermined number of turns (which is a multiple of the number of players), the first part of this game is over.
<br />
<br />In the second part of the game, the players take turns rewriting the entire list each turn with one digit the player chooses flipped from 0 to 1 or from 1 to 0.
<br />
<br />The new list cannot match any list previously arrived at during the game.
<br />
<br />If the lengths of the runs of 0's and 1's form a permutation of the lengths of the runs from any previous list, then the currently moving player gets a point.
<br />
<br />(It doesn't matter if a particular run-length, an element in the permutation, was for a run of 0's or for a run of 1's.)
<br />
<br />The game continues until either a player first achieves a predetermined score or until no more moves are possible.
<br />
<br />The player with the greatest score wins.
<br />
<br />
<br />Example game (to start):
<br />
<br />001110101 (start: 2,3,1,1,1,1)
<br />000110101 (3,2,1,1,1,1 point)
<br />000111101 (3,4,1,1)
<br />000011101 (4,3,1,1 point)
<br />100011101 (1,3,3,1,1)
<br />101011101 (1,1,1,1,3,1,1)
<br />111011101 (3,1,3,1,1 point)
<br />111010101 (3,1,1,1,1,1,1 point)
<br />111010111 (3,1,1,1,3 point)
<br />... etc.
<br />
<br />Thanks,
<br />Leroy Quet
<br />Amorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-42970848288409417582011-07-22T10:47:00.000-06:002011-07-22T10:48:31.380-06:00Combining And Dividing Integers GameHere is a game for any plural number of players.<br />(And no grids!)<br /><br />Players take turns, on each turn writing any positive integer to the end of a growing list of integers. The players do this until there is a predetermined number of integers in the list. (The number of integers is a multiple of the number of players.)<br /><br />Players then take turns; on each turn a player rewrites the list with one of these changes:<br />Either:<br />The player changes one integer in the list to a product of any number of PROPER divisors, each divisor greater than 1, that all multiply to that integer. (The divisors must be placed between the same integers in the list as the integer they replaced, but they can be in any order amongst themselves between those integers.)<br />Or:<br />The player replaces two adjacent integers in the list with their sum. (The sum goes in the same location within the list as the two integers it replaces.)<br /><br />The first player to get a list completely of primes is the winner.<br /><br />If a large predetermined number of moves have taken place, and no one has yet won, then the game is a tie.<br /><br /><br />Example: <br /><br />2 players:<br /><br />List to start:<br />2, 4, 10, 4, 1, 10<br />P1:<br />2, 4, 2, 5, 4, 1, 10<br />P2:<br />2, 4, 2, 5, 5, 10<br />P1:<br />2, 6, 5, 5, 10<br />P2:<br />2, 6, 5, 15<br />P1:<br />2, 6, 20<br />P2:<br />8, 20<br />P1:<br />8, 4, 5<br />P2:<br />12, 5<br />P1:<br />2, 3, 2, 5<br /><br />Player 1 wins.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-12182339064174700412011-06-12T05:59:00.000-06:002011-06-12T06:00:11.042-06:00Divisor Over Sums GameHere is another game that uses an n-by-n grid drawn on paper.<br /><br />This game is for 2 players.<br /><br />Players take turns placing 1,2,3,... n^2 into empty squares of the grid, one number each turn.<br />(So, player 1 places the odd numbers into the grid, and player 2 places the evens.)<br /><br /><br />Let H represent the collection of variables h, where each h is the sum of a HORIZONTALLY-adjacent pair of integers in the grid.<br /><br />Let V represent the collection of variables v, where each v is the sum of a VERTICALLY-adjacent pair of integers in the grid.<br /><br />(So, H and V each consist of n*(n-1) variables.)<br /><br />Player 1 then chooses a variable d which divides at least one of the h's.<br />Player 1 gets as a score:<br />d*(number of h's that are divisible by d)<br /><br />Player 2 then chooses a variable d' which divides at least one of the v's.<br />Player 2 gets as a score:<br />d'*(number of v's that are divisible by d')<br /><br /><br />The player with the largest score wins.<br /><br /><br />For example, let us say two poorly-playing players have this n=4 grid:<br /><br />08 09 01 10<br />03 11 16 07<br />02 14 15 05<br />04 12 06 13<br /><br />H =<br />(17,10,11, 14,27,23,<br />16,29,20, 16,18,19)<br /><br />Let us say that player 1 picks 4, then he gets 4*3 = 12 points.<br />(Because h's = 16, 20, and 16 are divisible by 4.)<br />Had he picked 29, he could have gotten 29, however.<br /><br /><br />V =<br />(11,5,6, 20,25,26,<br />17,31,21, 17,12,18)<br /><br />Player 2 picks 3 and gets 3*4 = 12 points.<br />She should have picked 31.<br /><br />Yes, you can always score at least n*(n-1) by picking d or d' = 1.<br /><br />As implied in the example, a player is responsible for finding his/her own score.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-76329643296728017502011-06-04T15:00:00.001-06:002011-06-04T15:00:44.376-06:00Palindromic Rows/ColumnsThis game is for 2 players.<br /><br />Start by drawing an n-by-n grid on paper.<br />(Who would have guessed??)<br /><br />The players take turns filling in empty squares of the grid, one square per move. Each player fills in floor(n^2/4) squares, so that there are a total of 2*floor(n^2/4) squares (about 1/2 of grid) filled in at game's end.<br /><br />Player 1 gets as a score the number of grid-squares whose states need to be changed (from filled to unfilled, or vice versa) in order to make each ROW a palindrome.<br /><br />Player 2 gets as a score the number of grid-squares whose states need to be changed in order to make each COLUMN a palindrome.<br /><br />The player with the SMALLEST score wins.<br /><br />{I know that the rules could have had Player 1 score with the number of squares needed to make the *columns* into palindromes, and Player 2 could have had the rows, then the *highest* score wins; but if each player attempts to determine their own score, then with my way each player has an incentive to be as efficient as possible in determining how many squares need to be changed to achieve the palindromes.}<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-90548059814851703422011-05-12T05:40:00.000-06:002011-05-13T14:47:08.609-06:00(Not)Divisors Or (Not)MultiplesThis game is for any plural number of players.<br /><br />Start with an n-by-n grid drawn on paper. (I suggest an n of at least 4 or 5 if there are two players who are beginners. I suggest a larger n if there are more players.)<br /><br />The game starts with Player 1 placing any integer >=2 in any square of the grid.<br />Thereafter, the players take turns. On a turn a player places any integer >= 2 in any empty square that is adjacent to and above, below, left of, or right of any square with an integer in it already.<br /><br />Each number must not have been used in the game yet.<br /><br />And each number written down must be a divisor or a multiple of each preexisting number that is orthogonally (above, below, left of, right of) adjacent to it.<br /><br />And each number written down must NOT be a divisor or a multiple of (may be coprime or not coprime to) each preexisting number that is diagonally adjacent to it.<br /><br />The last player able to move wins.<br /><br />Example finished game:<br />(n = 4. ** denotes an empty square. Leading 0's here are for formatting.)<br /><br />** ** ** 15<br />** ** 10 05<br />08 04 02 **<br />** ** 06 03<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-4670575166792661282011-05-07T06:57:00.001-06:002011-05-07T06:57:48.798-06:00Plexus Via Midpoints And EndpointsThis is a game for two players.<br /><br />Start with a lattice of n-by-n dots (which are the vertices of an (n-1)-by-(n-1)-squares grid). n is odd.<br /><br />One of the players draw a straight line-segment from the middle dot of the lattice to any adjacent dot above, below, left of, or right of the central dot.<br />(It doesn't matter which direction this line-segment is drawn.)<br /><br />The game consists of "full-moves", each of which consists of two "half-moves". In a full-move, one player moves then the other. The order of the players in the full-moves alternates.<br />So, we have the players moving like this:<br />(1,2),(2,1),(1,2),(2,1),(1,2),(2,1),...<br /><br />In a full-move, the players each draw a straight line-segment from a dot already connected with another dot via a line-segment, to an adjacent dot (in the direction of either above, below, left of, or right of). The adjacent dot must have no line-segments yet connected with it.<br /><br />The first player to move within a full-move decides if his/her line segment will be drawn from a dot already connected to 2 or 3 line-segments (a midpoint), or from a dot already connected to by exactly one line-segment (an endpoint).<br /><br />The second player to move in the full-move must also draw his/her line-segment from a midpoint or from an endpoint just as the first player in the full-move did.<br />In other words, if the first player in the full-move draws from a midpoint, the second player must also draw from a midpoint. And if the first player drew from an endpoint, the second player must also draw from an endpoint.<br />(A player is free to decide which midpoint or endpoint to draw from, however, provided that the dot he/she draws to doesn't have any line segments connecting to it yet.)<br /><br />If it is impossible for the second player in the full-move to do as the first player did (draw from midpoint or draw from endpoint), he/she loses his turn in this particular full-move, and the first player to move in that full-move then gets a point. (The second player still moves first in the next full-move, though.)<br /><br />The game continues until all dots have each been connected with at least one line-segment.<br /><br />The player with the largest score wins.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-55929703746006872572011-04-26T05:51:00.000-06:002011-04-26T05:52:30.464-06:00Palindromes/Antipalindromes In SequenceGame for 2 players.<br /><br />Make a row of m squares drawn on paper, where m is decided by both players.<br />(I suggest that m be >= 16 for beginners.)<br /><br />Players take turns. On a turn, a player places either a 0 or a 1 into any empty square.<br /><br />After a total of m moves, and when each square has one digit in it, play is over.<br /><br />Player 1 gets the sum of the lengths of all (not necessarily distinct but possibly overlapping*) palindromes<br />(a(1),a(2),a(3)...a(3),a(2),a(1))<br />of EVEN length (in the row of 0's and 1's) as her/his score.<br />Player 1's score =<br />sum{k=1 to [m/2]} 2k*(# of palindromes of length 2k).<br /><br />Player 2 gets the sum of the lengths of all (not necessarily distinct but possibly overlapping*) antipalindromes<br />(a(1),a(2),a(3)...1-a(3),1-a(2),1-a(1))<br />(of even length) (in the row of 0's and 1's) as her/his score.<br />Player 2's score =<br />sum{k=1 to [m/2]} 2k*(# of antipalindromes of length 2k).<br /><br />*(By "not necessarily distinct", I mean the patterns of 0's and 1's in different palindomes/antipalindromes may match. I of course don't mean the SAME palindrome/antipalindrome {in the same position and of the same length} can count more than once.)<br /><br />The player with the largest score is the winner.<br /><br />For example, let us say we have the following row (m=16):<br />1001001101000100<br /><br />We have the following palindromes of even length:<br />00, 00, 11, 00, 00, 00<br />1001, 1001, 0110<br /><br />So player 1 gets (2*6 + 4*3 =) 24 points.<br /><br />We have the following antipalindromes:<br />10, 01, 10, 01, 10, 01, 10, 01, 10<br />0011, 1010<br />100110, 110100<br />01001101<br /><br />So player 2 gets (2*9 + 4*2 + 6*2 + 1*8=) 46 points.<br /><br />Player 2 wins.<br /><br />(Did I make a mistake counting up the palindromes and antipalindromes in my example?)<br /><br />I suggest, if this game isn't played on a computer, that players count their own (anti)palindromes, and confirm them with their opponent. (If a player misses any that would contribute to his/her own score, it would be his/her own fault.)<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-61551792427062637172011-04-22T12:25:00.000-06:002011-04-22T12:26:10.051-06:00Divisors/Multiples Sequence GameHere is a game for any plural number of players.<br /><br />r is a positive integer agreed upon by all players before the start of the game.<br />I suggest that r be congruent to 1 mod {the number of players}.<br /><br />The game starts with a(1) = 1 and b(1) = 1. m and n both equal 1.<br /><br />Players alternate moves.<br /><br />(*)n=n+1.<br />(**)m=m+1.<br />A player on his move picks an integer a(m) that is either a divisor or a multiple of a(m-1).<br />a(m) must be <= r.<br />The player gets max(a(m),a(m-1))/min(a(m),a(m-1)) added to his score if a(m) is not among (b(1),b(2),...b(n-1)).<br />But the player gets<br />2*max(a(m),a(m-1))/min(a(m),a(m-1))<br />added to his score if a(m) is among (b(1),b(2),...b(n-1)).<br />The player continues his turn as far as he wants, but until a(m) is not amongst (b(1),b(2),...b(n-1)).<br />If the player is to continue his move, he goes to (**).<br />If a(m) is not among (b(1),b(2),...b(n-1)) and the player wants to end his move, then b(n) = a(m), and switch whose turn it is, go to (*) if n is < r.<br /><br />Play until n = r, and (b(1),b(2),...b(n)) is a permutation of the integers 1 through r.<br />The winner is then the player with the LOWEST score.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-71122434645755449032011-04-09T10:01:00.002-06:002011-04-10T06:40:27.953-06:00Sums Equal 1,2,3,...m In Small GridThis can be considered both a puzzle and a solitaire game.<br /><br />Make an n-by-n grid on paper. (I suggest that n = 3 for beginners.)<br /><br />Fill the grid with UNIQUE positive integers, one integer per square of the grid. The numbers need not be consecutively valued necessarily.<br /><br />Your score is the largest integer m such that integers 1 through m all occur as sums within the grid (without missing any positive integers <=m).<br /><br />A "sum" is of any number of addends (possibly just 1) that are all *consecutively placed* within a row of the grid or a column of the grid. (No diagonals in this variation.)<br /><br />So, for example, if we have the following 3-by-3 grid:<br /><br />2 9 8<br />3 1 10<br />7 6 15<br /><br />...the sums 1 through 19 all occur in this grid. So, I get a score of 19.<br />(Notice that some values of sums occur more than once.)<br /><br />I bet it is easy to do better than I did with a 3-by-3 grid, since my grid is inefficient, and I didn't try too hard to find it.<br /><br />(Note: There are 27 possible sums in a 3-by-3 grid of 1, 2, or 3 consecutively placed addends. I don't know what the largest possible score is for a 3-by-3 grid, though.)<br /><br />You can "play" someone else by both of you trying to score as well as you can on same-sized grids. Just try to outscore your opponent.<br /><br />Update: The largest score I personally received on the 3-by-3 grid is 20. But someone using a computer, I think, found a 3-by-3 grid with a score of 25. <br />Update2: Someone else found a 3-by-3 grid with a score of 26.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-9205077680362281962011-04-01T11:42:00.001-06:002011-04-01T11:42:58.227-06:00Multiples/Divisors Blob GameThis game is for 2 players.<br /><br />Start with an n-by-n grid drawn on paper. (n should be at least 8, I suggest.)<br /><br />The first player fills in any one of the grid's squares to start.<br /><br />Thereafter, players continue to take turns each filling in one empty square each turn.<br />After the first move, each square that is filled in must be immediately next to (in the direction of above, below, left of, or right of) at least one square that is already filled in.<br /><br />After both players have each filled in floor(n^2/4) squares (for 2*floor(n^2/4) squares filled in total), the game is over.<br /><br />Player 1 gets as a score the number of ROWS of the grid meeting this condition: Every run-length (of runs each of either all filled in squares or all empty squares) in that particular row is either a multiple or divisor of every other run length in that row.<br /><br />Player 2 gets as a score the number of COLUMNS of the grid meeting this condition: Every run-length (of runs each of either all filled in squares or all empty squares) in that particular column is either a multiple or divisor of every other run length in that column.<br /><br />The player with the largest score wins.<br /><br />Example: (n=12)<br /><br />o o * o o o o o o o o o<br />o o * * o o * * o o o o<br />o o o * * o * * o * * *<br />o o o o * * * * * * o *<br />o o o * * * * o o * * *<br />o o o * o o * o * * * o<br />o * * * * * * * o o * o<br />o * o o * * o * * o * *<br />* * o o * * * * o o * *<br />* o o o * o o * * o * o<br />* * o * * * o o o * * *<br />o o o o o * * o * * * o<br /><br />Run-lengths of rows:<br />(2,1,9)<br />(2,2,2,2,4) Point!<br />(3,2,1,2,1,3)<br />(4,6,1,1)<br />(3,4,2,3)<br />(3,1,2,1,1,3,1)<br />(1,7,2,1,1)<br />(1,1,2,2,1,2,1,2) Point!<br />(2,2,4,2,2) Point!<br />(1,3,1,2,2,1,1,1)<br />(2,1,3,3,3)<br />(5,2,1,3,1)<br /><br />Run-lengths of columns:<br />(8,3,1) <br />(6,3,1,1,1) Point!<br />(2,4,1,5)<br />(1,2,1,3,3,1,1)<br />(2,3,1,5,1)<br />(3,2,1,3,1,2)<br />(1,6,1,1,2,1) Point!<br />(1,3,2,4,2)<br />(3,1,1,1,1,1,1,1,1,1) Point!<br />(2,4,4,2) Point!<br />(2,1,1,8) Point!<br />(2,3,2,2,1,1,1)<br /><br />Player 2 wins, 5 to 3.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-20524540322525184152011-03-21T11:32:00.000-06:002011-03-21T11:33:07.851-06:00Sum Kind Of GameThis is a game for two players.<br />Start with an n-by-n grid drawn on paper, where n is even.<br /><br />Players take turns placing numbers into the empty grid squares, one number per move.<br />Player 1 places into the grid a 1 then a 2 then 3 then... then the value of n^2/2-1 then finally the value of n^2/2.<br />Player 2, on the other hand, goes the opposite way, placing into the grid the value of n^2/2 then the value of n^2/2-1 then... then 3 then a 2 and finally a 1.<br /><br />Player 1 gets as a score the number of squares with integers S where each is the sum of the values in two different squares of the same ROW (same row as their sum S).<br /><br />Player 2 gets as a score the number of squares with integers S where each is the sum of the values in two different squares of the same COLUMN (same column as their sum S).<br /><br />(The addends can be the same value, but must be in different squares.)<br /><br />It doesn't matter who wrote down each number as far as scoring is concerned.<br /><br />Largest score wins.<br /><br />Sample game: (n=6)<br /><br />01 02 03 04 05 18<br />12 17 05 15 11 10<br />10 14 13 17 03 06<br />09 14 06 04 09 02<br />08 16 07 15 08 12<br />16 18 07 13 11 01<br /><br />Player 1 gets for each row (top to bottom):<br />3, 2, 2, 1, 2, 1<br />11 points. <br /><br />Player 2 gets for each column (left to right):<br />2, 2, 1, 1, 3, 2<br />11 points.<br /><br />(I may have made a mistake or two figuring these out.)<br /><br />A tie, if I didn't err.<br /><br />----<br /><br />Is there a bias in favor of one player in the game because strategies are different due to the two orders the numbers are placed into the grid?<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-18239915462788286842011-03-12T12:28:00.001-07:002011-03-12T12:28:59.298-07:00Flipping Bits/SquaresThis game is for any plural number of players.<br /><br />You need a blank piece of paper and a pen/pencil.<br /><br />A predetermined number of rounds are played.<br /><br />Start each round by making a row of n squares, where n is at least 7 or more if there are 2 players; n is larger if there are more players.<br />Fill in every other square.<br /><br />Take turns.<br /><br />On a turn a player can change any ONE square from filled in to not filled in, or vice versa.<br />Make a new row of squares to reflect the move.<br /><br />A move may not lead to a pattern of how the squares are filled that has already existed in the round.<br />If all patterns achievable by changing one square (flipping one bit) lead to patterns that have already existed in the round, then a player flips two bits. If all patterns achievable by flipping two bits also lead to patterns that already occurred in the round, then flip three bits; etc.<br />A player MUST move with the fewest number of bits flipped as possible that will lead to a new pattern for the round.<br /><br />As soon as a player achieves with their move a pattern with exactly one bit/square a different color (either filled or not filled) than the rest of the row, then that player gets {the number of squares from the left side of the row that is this unique square's position} added to his/her score.<br /><br />The round is then over.<br />(So, one player scores each round.)<br /><br />Play a predetermined number of rounds, adding up each player's scores.<br /><br />The player with the largest grand total wins.<br /><br /><br />Example round: (n = 7)<br /><br />(*)( )(*)( )(*)( )(*): start<br />(*)( )( )( )(*)( )(*): player 1<br />(*)( )( )( )(*)( )( ): player 2<br />(*)( )( )(*)(*)( )( ): player 1<br />(*)(*)( )(*)(*)( )( ): player 2<br />(*)(*)( )(*)(*)( )(*): player 1<br />(*)(*)(*)(*)(*)( )(*): player 2<br />Player 2 gets 6 points, since the blank square is the 6th square from the left.<br /><br />(I wonder what the funnest n is for any given number of players, especially for 2 players.)<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-16235712714186399562011-02-22T10:48:00.001-07:002011-02-22T10:48:57.718-07:00Another Binary Primes GameThis is a game for 2 players.<br /><br />Draw a single column of n squares.<br /><br />Players take turns. On a turn a player writes an x into any blank square. Each player makes a total of floor(n/4) x's, so about half the squares are filled at game's end.<br /><br />A player's score equals the number of DISTINCT primes they can find whose binary representation is a substring in the columns, if each x is interpreted as a 1 and each blank square is interpreted as a 0. Player 1 finds binary primes written from top to bottom in the column, and player 2 finds primes written from bottom to top in the column.<br /><br />Largest score wins.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-50512034930356310192011-01-15T06:32:00.001-07:002011-01-15T06:32:43.698-07:00Binary Primes GameThis is a game for two players.<br /><br />Start by drawing an n-by-n grid on paper. (I don't suggest that n be too big, unless you are playing this game on a computer.)<br />n is odd. (Thanks to Ilmari Karonen for pointing out that a win can always be forced by player 2 if n is even {except for a likely small number of even exceptions}.)<br /><br />The players take turn filling in squares, each player filling in one empty square per move.<br />Each player fills in floor(n^2/4) squares total.<br />(So, 2*floor(n^2/4) squares are filled in all together at the game's completion.)<br /><br /><br />Now, interpret the filled-in (black) squares and the blank (white) squares in each row and column as the digits of a binary number. In any particular row or column, either all the black squares represent a 0 or all of the black squares represent 1. And the white squares each equal the opposite binary digit than the black squares represent in that row/column. And the binary number can be read either top to bottom, or bottom to top (for each column), or left to right, or right to left (for each row).<br /><br />So, to be clear, the binary digit (0 or 1) represented by a color has to remain the same within any particular row or column, but can differ between different rows and columns.<br /><br />And the direction the binary number is read can differ between different rows and columns.<br /><br />So, after the game is complete, the players go through and write down in two lists, one for the columns and one for the rows, the decimal representation of the largest possible PRIME possible, if any is possible, for each row and each column.<br /><br />Player 1 gets as a score the product of all of the primes in the columns.<br />Player 2 gets as a score the product of all of the primes in the rows.<br /><br />The player with the largest score wins.<br /><br />Example (randomly "played" with no strategy):<br />n=6:<br /><br />. * * . . * <br />* * . . . *<br />. * * * . .<br />. * * . . .<br />* . * * * .<br />. * . . * *<br /><br />Largest primes, or 1 if all possibilities are composite or 1:<br /><br />For columns:<br />1, 61, 29, 53, 3, 1.<br />Score =<br />61*29*53*3 = 281271.<br /><br />For rows:<br />1, 1, 1, 1, 29, 19.<br />29*19 = 551.<br /><br />Player 1 (columns) wins, obviously.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com2tag:blogger.com,1999:blog-2237244637228809591.post-28387553088528150862010-12-21T04:45:00.001-07:002010-12-21T04:45:33.775-07:00Permudrome -- Grid GameHere is a game for 2 players.<br />The game's name is a combination of the words "permutation" and "palindrome".<br /><br />Start with an n-by-n grid,<br />where n is a multiple of 4.<br />I suggest that n is >= 8.<br /><br />The players take turns. On a turn a player draws two x's into the grid, each x into an empty square such that no column or row has more than one x.<br /><br />After there is exactly one x in each row and in each column -- n x's total, n/4 moves for each player -- play is over.<br /><br />Write down the (n-1) absolute values in order, of the changes in the vertical positions of adjacent x's from column to column, along the bottom of the grid.<br />Write down the (n-1) absolute values in order, of the changes in the horizontal positions of adjacent x's from row to row, along the left side of the grid.<br /><br />Player 1 gets as a score the length of the largest palindromic subsequence within the sequence of vertical changes written along the bottom of the grid.<br /><br />Player 2 gets as a score the length of the largest palindromic subsequence within the sequence of horizontal changes written along the left side of the grid.<br /><br />Largest score wins. (Ties are possible.)<br /><br />Example: n=12:<br /><br />. . x . . . . . . . . .<br />. . . . x . . . . . . .<br />. . . . . . x . . . . .<br />x . . . . . . . . . . .<br />. . . . . x . . . . . .<br />. . . . . . . x . . . .<br />. x . . . . . . . . . .<br />. . . x . . . . . . . .<br />. . . . . . . . x . . .<br />. . . . . . . . . . . x<br />. . . . . . . . . . x .<br />. . . . . . . . . x . .<br /><br />Changes in vertical positions column to column:<br />3,6,7,6,3,2,3,3,3,1,1<br />The largest palindromic subsequence is (3,6,7,6,3). Player 1 gets 5 points.<br /><br />Changes in horizontal positions row to row:<br />2,2,6,5,2,6,2,5,3,1,1,<br />The largest palindromic subsequence is (5,2,6,2,5). Player 2 gets 5 points.<br /><br />It is a tie.<br /><br />What about strategies for this game?<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-10206400931315680352010-12-13T05:48:00.002-07:002010-12-13T05:49:18.840-07:00One x TwiceFor 2 players.<br />Start with an n-by-n grid drawn on paper.<br /><br />A move consists of both players each secretly picking an integer between 1 and n.<br />Both numbers are then revealed. An x is then drawn in the grid-square that has the column number of player 1's number, and has the row number of player 2's number.<br />So, in other words, player 1 picks the horizontal position of the number, and player 2 picks the vertical position.<br /><br />If the x lands in an empty square, then the game continues.<br /><br />But, however, the first time an x lands in a square that already has an x, then the game is over. Player 1 wins if this final x was written on an oddly numbered move. Player 2 wins if this x was written on an evenly numbered move.<br />So, in other words, if there are an odd number of x's at game's end -- and an even number of squares with x's -- then player 1 wins. If there are an even number of x's -- and an odd number of squares with x's -- then player 2 wins.<br /><br />What kind of strategies will help you win at this game (if you cannot read the other player's mind)?<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-27451092895923491592010-12-13T05:48:00.001-07:002010-12-13T05:48:29.291-07:00Line & Unobscured Dots GameHere is an unoriginal game for 2 players.<br /><br />You need a blank piece of paper and a pen/pencil, maybe 2 pens/pencils of different colors.<br /><br />To start, someone draw a dot in the middle of the piece of paper. Then each player draws a different dot on the paper. (So, you have 3 dots.)<br /><br /><br />Thereafter, the players take turns forming a line of connected straight line-segments on the paper, plus drawing dots. On their turn, a player draws a straight line segment from THEIR END (of their color, if the players are using differently colored pens/pencils) of the connected string of line-segments (or from the central dot if this is the player's first time drawing a line-segment during the game) to any undrawn-to dot (a dot without a line-segment connected to it), such that the line-segment doesn't pass through any other line-segments or through any other dots along the way.<br /><br />Then, on the same move, the player draws 2 dots, neither on a line or on another dot. One dot is "visible" by the player's own end of the line of connected line-segments. The other dot is visible by the other player's end of the line of connected line-segments.<br /><br />By 2 points being "visible" to each other, it is meant that it is possible to draw a straight line-segment between the two points, and that line-segment doesn't pass through any intervening dots or lines.<br /><br />After a fixed number of moves, the same number of moves for both players, the game is over.<br /><br />The winner has, at game's end, the most number of undrawn-to dots visible from their end-point of the line of connected line-segments.<br /><br />Note: If playing with 2 differently colored pens/pencils, it doesn't matter what color the dots are. The only reason for using 2 different colors is to make it easier to see whose end of the line of connected line-segments is whose.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-16379575098358594912010-11-27T06:24:00.000-07:002010-11-27T06:26:48.597-07:00x's and lines(Is this game original? I myself may have come up with something similar earlier.)<br /><br />This is a game for 2 players.<br /><br />You have an n-by-n grid drawn on paper (as almost always).<br />n should be >= 8, I suggest.<br /><br />In the first part of the game, the players take turns placing a total of n x's into the grid, where one x is placed in an empty square of the grid each move.<br /><br />In the second part of the game, players take turns. On a turn, a player draws a line (through the centers of the intervening squares) either up, right, down, or left from the last x drawn to by the other player. The (maybe bending) line may take at most one right-angle turn. And it must end at an x not drawn to/from yet. (Player 1 draws from any of the x's on her/his first move.) (What I mean by "the line may take at most one right angle turn" is that the x's will either be connected by a single straight line-segment {if both x's are in the same row or column} or they will be connected by two perpendicular, connected straight line-segments {if the x's are in both a different row and a different column}).<br /><br />Lines can't pass through a line already drawn in the second part of the game. And the line cannot pass through an x on its way between two other x's during a move. The line may, though, share a corner with, or coincide partially with (both line-segments horizontal or both line-segments vertical), a line drawn previously during the second part of the game.<br /><br />A player must move if it is possible.<br /><br />The last player able to move LOSES.<br /><br />(Note: Unlike some other games I have posted recently, more than one x or no x's at all may be in any row or column.)<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-34147302816363728862010-11-09T05:16:00.001-07:002010-11-09T05:29:32.056-07:00Multiplications Within The PermutationThis is a game for any plural number of players. Let the number of players be m.<br /><br />Start with an n-by-n grid drawn on paper, where n = k*m + 1, k is some integer >= 3.<br /><br />In the first part of the game, players take turns placing x's in empty squares of the grid, one x per turn, such that no more than one x is in each row and in each column of the grid.<br /><br />After exactly n x's are placed in the grid, the first part of the game is over.<br /><br />The second part of the game starts with a 1 being placed in the leftmost square with an x in it. Players then take turns. <br />On the jth move (starting at move # 1) of the second part of the game, the moving player places a (j+1) in any square with an x and without a number already in it.<br />He/she gets added to her/his score:<br />|x(j)-x(j+1)| * |y(j)-y(j+1)|,<br />where x(j) is the number of squares from the bottom of the grid where the square with the j in it is located, and y(j) is the number of squares from the left side of the grid where the square with the j in it is located.<br /><br />So, what we are adding to the moving player's score (the score of the player writing a j+1 in a square) is the product of {the change in horizontal distance between the squares with j and j+1 in them} and {the change of vertical distance between the squares with j and j+1 in them}.<br /><br />When the nth x is numbered with a n= m*k+1, the game is over.<br /><br />Largest score wins.<br /><br />Here is an example:<br />n=7. m =2.<br />. 3 . . . . .<br />. . 6 . . . .<br />. . . . . . 5<br />. . . . 7 . .<br />1 . . . . . .<br />. . . 4 . . .<br />. . . . . 2 .<br /><br />Squares 1 to 2: 5*2 = 10<br />Squares 2 to 3: 4*6 = 24<br />Squares 3 to 4: 2*5 = 10<br />Squares 4 to 5: 3*3 = 9<br />Squares 5 to 6: 4*1 = 4<br />Squares 6 to 7: 2*2 = 4<br /><br />Player 1 gets: 10+10+4 = 24 points.<br />Player 2 gets: 24 + 9 + 4 = 37 points.<br />Player 2 wins.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-72782814369660875552010-10-30T07:34:00.000-06:002010-10-30T07:35:18.534-06:00Prime Target GameA game for any plural number of players: (Number of players = m.)<br /><br />Draw m*k+1 incrementally larger concentric circles on a piece of paper, where k is some positive integer >= 2.<br /><br />Subdivide the circles by drawing n equally spaced rays from their center, where n is at least 6, say.<br /><br />(So now you should have a target.)<br /><br /><br />The players start the game by taking turns, and each player on a turn places an integer -- 1 to n and which has not been written down earlier in the game -- in an empty pie-shaped wedge in the central circle. After n moves, there should be a permutation of (1,2,3,...,n) in the central circle, one integer per wedge.<br /><br />In the second part of the game, the players take turns, each player "completing a ring" on a move. By completing a ring, the player fills in the n sections of the innermost *empty* ring. The player fills in each section of the ring either with the sum of (the integer immediately adjacent to the section, but in the next ring inward) and (the integer one position clockwise to the section, but in the next ring inward), or with the absolute value of the difference between these particular two integers (in the next ring inward).<br /><br />Example:<br />\...8..|....../<br />-\-----|-----/-<br />..\.2..|..6./<br />---\---|---/---<br />8 = 2+6.<br />(The 8 could have been a 4.)<br /><br />After the ring is completed, the player gets the number of primes in his latest ring added to his score, OR, if there is exactly one prime in his ring (no more, no fewer), he gets the value of that prime added to his score.<br /><br />After all rings are completed, the game is over. Largest score wins.<br /><br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-28237009711289026892010-10-11T13:20:00.000-06:002010-10-11T13:21:19.969-06:00Grid Game Of DifferencesThis is a game for two player.<br /><br />Draw an n-by-n grid on paper, where I suggest that n is at least 8.<br /><br />The players take turns placing x's in the empty squares of the grid, one x per turn.<br /><br />No two or more x's may be placed in the same row or in the same column of the grid.<br /><br />After n total moves (when there is exactly one x in each row and column), the game is over.<br /><br />Now to determine the score:<br /><br />Reading left to right, write down the (n-1) absolute values of the differences between the consecutive x's' vertical coordinates, in terms of number of squares.<br /><br />In another list, reading bottom to top, write down the (n-1) absolute values of the differences between the consecutive x's' horizontal coordinates, in terms of number of squares.<br /><br />Player 1 gets a point for every distinct numerical value occurring in the first list of differences.<br />Player 2 gets a point for every distinct numerical value occurring in the second list.<br />If a particular difference occurs at least once in a single list, then the player gets one point for that particular difference.<br /><br />Largest score wins.<br /><br />We may need an example here:<br /><br />n=9:<br /><br />. x . . . . . . .<br />. . . . . . . . x<br />. . x . . . . . .<br />. . . . . x . . .<br />x . . . . . . . .<br />. . . x . . . . .<br />. . . . . . . x .<br />. . . . . . x . .<br />. . . . x . . . .<br /><br />Player 1's (vertical) differences (reading left to right) are:<br />4,2,3,3,5,4,1,5<br />The unique values that occur are:<br />1,2,3,4,5<br />Player 1 gets 5 points.<br /><br />Player 2's (horizontal) differences (reading bottom to top) are:<br />2,1,4,3,5,3,6,7<br />The unique values that occur are:<br />1,2,3,4,5,6,7<br />Player 2 gets 7 points.<br /><br />In another variation of this game, count ONLY those differences that occur exactly once (and no more than once).<br />In this variation, player 1 would have gotten 2 points, for the differences 1 and 2.<br />Player 2 would have gotten 6 points, for the differences 1,2,4,5,6,7.<br />(Since 3 is the only difference in this list that occurs more than once.)<br /><br /><br />Which variation is more fun?<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-21206075115871686962010-10-06T05:27:00.000-06:002010-10-06T05:28:02.843-06:00Bouncing Pathways Within A Circle: GameA game for two players:<br /><br />First, draw a circle on a piece of paper.<br /><br />Players start by each drawing a different straight line-segment at any angle they choose from the center of the circle to the circumference.<br /><br />Players thereafter move like so: (Player 2, Player 1), (Player 1, Player 2), (Player 2, Player 1), (Pl 1, Pl 2), (Pl 2, Pl 1), etc.<br />So, we have "whole moves", consisting of two moves, with a move by each player. And who moves first in the whole moves alternates.<br /><br />The first player to move in a whole-move decides if the next line-segment will bounce left or bounce right. This player then draws his straight line-segment in the proper direction (relative to the direction his own last line-segment was traveling) from where his own last line segment ended to where the new line-segment comes up against a pre-existing line-segment (drawn by either player) or up against the circumference of the circle. A player's line-segment may pass through a pre-existing line-segment. But each time a player crosses a line-segment with another line-segment, his score is halved. No line-segments may pass outside of the circle.<br /><br /><br />The second player to move in a full-move then must bounce the same direction, left or right, as the other player did, but relative to the direction this player's own last segment was traveling. And he draws his segment from where his own last line-segment ended to where his new line-segment comes up against another pre-existing segment or up against the circle's circumference. Again, his segment may pass through a pre-existing line-segment (but not pass through the circle's circumference), but doing so halves his score each time he does it.<br /><br />After a predetermined number of full-moves (such as 10), each player's score = the length of that player's final line-segment divided by 2^(the number of lines crossed by that player).<br /><br />Largest score wins.<br /><br />Note: To be clear, there will be two "pathways" within the circle: One pathway belonging to each player, and each pathway made up of the series of connected line-segments drawn by that player.<br /><br />Also, line-segments may not coincide, except at the points where they intersect.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-44090054497460611202010-09-18T06:02:00.001-06:002010-09-18T06:02:58.407-06:00Lengths Of Lengths Of Lengths GameThis game is for any plural number of players.<br /><br />There are a predetermined number of rounds. Each player plays the same number of rounds as the offense player.<br /><br />In a round: A list is made of 0's and 1's, starting at empty-set. The players take turns each appending either a 0 or a 1 to the end (right side) of the list. The total number of digits in the round's list is n, where n is a predetermined number that is a multiple of the number of players.<br />(n is the same for all rounds.)<br /><br />After the list of 0's or 1's is made, we determine the score.<br /><br />We now form a series of lists.<br /><br />(*) If the new list consists entirely of 1's, then the offense player gets the number of 1's added to his/her score. And the round is over. (If the round is over, then go to **.)<br /><br />If there is at least one number not equal to 1 in the latest list: Below the last list made, write a new list consisting, in order, of the lengths of the runs of similarly-valued numbers from the previous list.<br /><br />Go to (*).<br /><br />(**) Change who is the offense player. Start a new round.<br /><br />After all rounds have been played, the player with the largest score wins.<br /><br />Sample round: n = 25:<br /><br />1010110011101000111001011<br /><br />1,1,1,1,2,2,3,1,1,3,3,2,1,1,2<br /><br />4, 2,1,2,2,1,2,1<br /><br />1,1,1,2,1,1,1<br /><br />3,1,3<br /><br />1,1,1<br /><br />Offense gets 3 points.<br /><br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0tag:blogger.com,1999:blog-2237244637228809591.post-49028415899624945022010-09-08T13:25:00.000-06:002010-09-08T13:26:34.333-06:00Plusses And Minuses Grid GameThis game is for two players.<br /><br />Start with an n-by-n grid drawn on paper. I suggest that n be an odd integer >= 9.<br /><br />The first part of the game doesn't involve the grid. In this part of the game, the players take turns each contributing to their own "prediction list" of +'s and -'a. On each move, a player appends to the end of their list either a + or a -. Each player is aware of the other player's list as it is being made. After both players' lists are n-1 symbols long, this part of the game is over. (See below for the significance of the lists.)<br /><br />I suggest at this point that a dot be put at the lower left corner of the grid so as to keep this corner straight from the others.<br /><br />In the next part of the game, the players take turns, each move filling in an empty square of the grid. The filled in square must not be in the same row or column as any other filled in square.<br /><br />(I suggest that the player who moved first in the first part of the game moves second in the second part of the game. Just to be fair.)<br /><br />After n squares total have been filled in, this part of the game is over.<br />(There will be exactly one filled in square in each row and in each column.)<br /><br />Now, we determine the scores.<br />Take the grid, with the dot in the lower left corner.<br />Going from the left to the right, the filled in squares represent a permutation P = (p(1),p(2),...p(n)) of (1,2,3,...,n). Player 1 forms a "truth list" of +'s and -'s, where the kth symbol is the sign of p(k+1)-p(k).<br /><br />Going from bottom to the top, the filled in squares represent a permutation Q = (q(1),q(2),...q(n)) of (1,2,3,...,n), where Q is the inverse permutation of P. Player 2 forms a truth list of +'s and -'s, where the kth symbol is the sign of q(k+1)-q(k).<br /><br />(The lower-left square represents 1 both in respect to permutation P and permutation Q. And the upper right square represents n for both permutations.)<br /><br />Write each players truth list below their prediction list so that respective signs are lined up. Each player gets a point for each corresponding pair of signs that match.<br /><br />Largest score wins.<br /><br />Here is a sample game:<br /><br />n = 8.<br /><br />Grid:<br />. * . . . . . .<br />. . . . . . . *<br />. . . . . . * .<br />. . . . . * . .<br />. . * . . . . .<br />. . . . * . . .<br />. . . * . . . .<br />* . . . . . . .<br /><br />Player 1's prediction list:<br />+ - + - + - +<br />Player 1's truth list:<br />+ - - + + + +<br />The first pair, second pair, 5th pair, and 7th pair match. So, player 1 gets 4 points.<br /><br />Player 2's prediction list:<br />+ + - + + - +<br />Player 2's truth list:<br />+ + - + + + -<br />The first pair, second pair, 3rd pair, 4th pair, and 5th pair match. So, player 2 gets 5 points.<br /><br />Player 2 wins.<br /><br />Thanks,<br />Leroy QuetAmorphous Trapezoidhttp://www.blogger.com/profile/13848496983638628005noreply@blogger.com0