Thursday, February 26, 2009

Words, Letters, & Logic

This is a word game inspired by the mathematical game at the link below:
(But this game should be more fun for the anti-math crowd.)

This game, which is for 2 players, is sort of like a cross between Scrabble and Sudoku. Sort of.

The game is divided into several phases.
In the first phase the players take turns writing letters into the squares of a 4-by-4 grid (or a 5-by-5 grid for more advanced players).
Each letter should appear at most once in the grid.

So, for example, we could have this grid:


Then, in the next phase the players take turns coming up with a list of words associated with each letter of the grid. Let the letter associated with a word be a "grid-letter" (because the letter appears in the grid). A word or phrase (that may be almost nonsense, if no shorter words can be thought of) must contain all of the letters in the squares immediately adjacent (in the directions of above, below, left of, and right of) to the word's grid-letter, but the word need not contain the grid-letter itself.

The list of words, each word written next to its grid-letter, is ordered by the grid-letters in alphabetical order.

So, in my example we can have the list of words (written right of their grid-letters):

A: mocked
B: blighted
C: dark gods (Notice that this is an arbitrary-sounding phrase.)
D: backed
E: mined
G: backpacker
H: pinto bean
I: bean
K: lack
L: sky
M: rake
N: high
P: quag hole (Another arbitrary phrase. A quag is a marshy place.)
Q: pray
R: quagmires
S: clear

For example, the letter A in the grid is next to D, K, and M. And the word "mocked" contains these letters.
(Note: Words with lots of letters are more likely to make the game easier for both players. Words with fewer letters are more likely to make the game harder for both players.)

In the next phase the players each draw their own empty 4-by-4 (or 5-by-5) grid.
Then each player writes any one letter that occurs in the original grid into any square of their OPPONENT'S grid.

Then the original grid of letters is hidden. (So the list of words should be drawn on a different piece of paper than the original grid of letters.)

In the final phase the players try to each fill their own grid, given the letter written in their grid by their opponent, with the same letters that were in the original grid (one letter per square of the grid), never repeating a letter (including the one letter written by the player's opponent in the player's grid), such that all letters adjacent (in the directions of above, below, right of, and left of) to a letter in the grid occur in the word associated with that grid-letter.

Remember that if two letters are adjacent (say, letter 1 and letter 2), then letter 1 must be in letter 2's word AND letter 2 must be in letter 1's word.

The players each try to fill in as many squares as they can under the rules.
Their score is the number of squares they correctly fill in with letters.
(Note: Not all grids can be filled in completely. It depends on where a player's opponent places the first letter in the player's grid.)

If a player makes a mistake (a letter doesn't appear in one of its adjacent letter's words, a letter is written in a player's grid that wasn't in the original grid, or a letter occurs more than once in a player's grid), than that player forfeits.

If neither player forfeits, then the winner is the player who filled in the most number of squares in their own grid.
(Ties possible.)

Back to the example, here is a player's grid with an E put in the lower left square by the player's opponent:
(* is an empty square.)

R * * *

13 points.

(Note: I couldn't put a G in the empty square at position (2,2) because there is no G in "mocked".)

Leroy Quet

Sunday, February 22, 2009

Arranging Numbers By Rules -- A Game Also A Puzzle

This seems like it would be a fun game.

This game is for 2 players. Start by drawing an n-by-n grid on a piece of paper, where n is at least 4 or 5 (but not too massive). I suggest that n be even (to make this game fair for both players).

First the players take turns placing the integers 1 through n^2 into the grid's squares so that there ends up being exactly one integer in each square of the grid.
Player 1 places the odd integers in the grid's squares, and player 2 places the even integers.

Then the players take turns making up rules or classifications, one rule per each integer from 1 to n^2, where each rule defines a class of integers which includes all the integers immediately adjacent (in the directions of above, below, left of, and right of) to the integer which matches the number of the rule.
(The rule need not match the number of the rule itself.)
In other words, say that we are concerned with the rule defining the neighbors of the integer 3 in the grid. Left of the 3 happens to be, in this example, a 5. Above the 3 happens to be a 9. Below the 3 is a 2. And right of the 3 is a 17.
So, the neighbors of the 3 are 5, 9, 2, 17. There are of course an infinite number of classes that these number fall into. But one of the classes is (2^k + 1), since all 4 of the integers are 1 more than a power of 2. So rule #3 could be "Numbers of the form (2^k +1), k >= 0".

So, player 1 makes up the rules for the neighbors of the odd integers. And player 2 makes up the rules for the neighbors of the even integers.

I encourage players to be creative when coming up with rules. Yes, a rule could look like: "One of these integers: 2,6,5,9", or on the other extreme: "Any integer at all". But making a rule too broad or too narrow affects both players equally.

Next, after the rules are constructed, each player draws an empty n-by-n grid for themselves. Each player then places into any square of their opponent's grid any integer from 1 to n^2.

Then, each player tries to fill in the remaining squares of his/her own grid so that, given the integer her/his opponent already placed in her/his grid, each integer's immediate neighbors (in the direction of above, below, right of, left of)
follows the corresponding rule for that integer. The original grid of numbers is hidden while the players each try to solve the puzzle.

Remember that if two integers, j and k, are adjacent, then not only does j have to follow rule #k, but k has to follow rule #j as well.

A player's score is the number of squares he/she fills in successfully. If a player makes a mistake (a number doesn't follow the rule for the number it is adjacent to, or a specific number appears more than once in a player's grid, or a number in the player's grid is not an integer >= 1 and <= n^2), then the player forfeits.

I also should make an addition to how the game is played.
Rules such as rule 7 below, where other squares' values have something to do with the rule, can be problematic if not all the relevant squares are filled in.
So, if rule #m says that the values of the neighbors of the m must depend on other square's values in some way (the other squares which rule #m depends upon we call S), then the squares of S must all be filled with numbers, IF any of the neighbors of square m are filled, or else the player forfeits.

In my example, all the numbers in the same row as the 7 must be filled, if any of the numbers adjacent to the 7 are filled, or else I would automatically lose to my opponent (unless we both forfeited for any reason).


Here is a sample 4-by-4 grid with rules.

01 04 09 16
02 03 08 15
05 06 07 14
10 11 12 13

1: Power of 2.
2: Prime power.
3: Even.
4: Power of 3.
5: Squarefree.
6: Prime.
7: Not coprime with the sum of the integers (including the 7) in the same row as the 7.
8: Odd.
9: Power of 2.
10: One less or one more than a triangular number.
11: Prime - 1.
12: Prime.
13: Between 10 and 15 (inclusive).
14: Divides 14 or is coprime to it.
15: Squarefree integer + 1.
16: Multiple of 3.

Let us say, without me trying to actually solve the puzzle, that the player's opponent puts a 3 in the lower left square of an empty 4-by-4 grid. How many integers can be put in this grid correctly under the rules?

Note: Any integer (from 1 to n^2, not occurring anywhere else in a player's grid) can occur anywhere in a player's grid where there would be no integers in the neighboring squares immediately above, below, right of, or left of it, of course. (No rules are violated here.)

Leroy Quet

Tuesday, February 17, 2009

Primes, Moves, & Motions

This is a game for 2 players. It is played on an n-by-n grid, where n is at least 8 or higher, I suggest.

Player 1 starts the game by placing a 1 in any square of the grid.

The game consists of "moves" alternately taken by each player. Each move is made up of a series of "motions", where a single player makes all of the motions in any particular move.

A player on move n (where player 1 placing the 1 any square is move #1 and is motion #1) makes motions p(n-1) through p(n)-1 (for moves n>=2), where p(n) is the nth prime.
Player 1 makes the odd numbered moves, while player 2 makes the even numbered moves.

On MOTION m, a player places the number m in any EMPTY square that is adjacent to the square with a (m-1) in it (which was placed in the (m-1) square by either player), such that:

*If m is an even composite, the player places m immediately either left of or right of the square with an (m-1) in it.
*If m is an odd composite, the player places m immediately either above or below the square with an (m-1) in it.
* If m is a prime (ie. If this is the first motion of a player's move), then the player can place m in the square that is immediately either above, below, right of, left of, or diagonal to the square with the (m-1) in it.

The last player that can make a motion loses.

Variation: The first player that cannot make a motion loses.
(The difference between the original version and the variation is simply that in the original version, if a player places a p-1, where p is a prime, but the other player can't place a p, then the player who placed the p-1 loses. In the variation, the player who cannot place a p loses.)
I leave it up to players to decide amongst themselves which version they prefer.

Leroy Quet

Thursday, February 12, 2009

Draw Lines And Shade Sections

(I have posted other games before where you draw lines, then shade in sections bordered by the lines. But I can't think of a better name for this game.)

This game is for any plural number of players.

Start with an n-by-n grid lightly drawn on paper.

Players take turns drawing horizontal and vertical line segments, each segment being one grid-square side in length, from grid-vertex to adjacent vertex along the lightly drawn lines of the grid.
Player 1 starts the game by drawing a line segment from any vertex to adjacent vertex. Players each, thereafter, draw a line segment from where the last line segment left off. Darker line segments must not be drawn where other darker line segments were previously drawn. And the continuous path of line-segments must not cross itself. Yet, the path may be drawn to any vertex more than once.
This part of the game continues until the path cannot be drawn anymore.
(If all players want an interesting game, they should probably try not to lead the path into a situation where it prematurely ends.)

After the darker path of line-segments is complete, the players then take turns filling in un-shaded-in sections of the grid, one section per move. By "section", I mean a polygon bounded by the darker line-segments, or by vertexes where different parts of the path come together, or by the perimeter of the grid. (You can draw darker line segments along the border of the grid. But as far as the sections and the border of the grid are concerned, whether a particular segment of the border of the grid was darkened in or not does not matter.)

As soon as a player is forced to fill in a section bordering (along a line) another filled in section, or accidently does so, then that player is removed from play.
(Two filled in sections may border at a vertex without removing a player.)

Play continues until there is one player left, who then is the winner.

By the way, I suggest that when two parts of the path come together at a single vertex, then the path should be drawn so it is clear that there is no gap between the parts of the path. Otherwise, players may think that two of the sections that meet at that vertex are only one section (with a choke-point).

Leroy Quet

PS: Don't get confused between "sections" and "segments".

Thursday, February 5, 2009

Squares Overlapping/ Subdividing Into Primes

Here is a game for any number of players.

Play a number of rounds, where the number of rounds is a multiple of the number of players. Each player plays the same number of rounds as offense.

Start each round with with an n-by-n grid lightly drawn on paper, where n is at least 15 or more and is finite. (And n is the same value for all rounds.)

On a round the players take turns boldly drawing the perimeters of squares along the lines of the grid. The edges of the squares may overlap. But each player must darken in at least some segment(s) of grid-lines that have not yet been part of any edge of any previously drawn square.

After a predetermined number of moves (the same number for all rounds), then the offense player tries to find the largest CONTIGUOUS collection of regions bordered by the bold lines that have a total of a prime number of grid-squares in them.
(The regions can be of any shape, and may be made by adding squares together or by taking smaller squares out of larger regions, for example.)

The offense player gets this prime added to their score.

After all rounds are played, the player with the highest total score wins.

Leroy Quet