Re: Stepping By Divisors -- Grid Game
- From: Don Woods <don-ns@xxxxxxxxxx>
- Date: 18 Mar 2009 22:43:03 -0700
I'll start by suggesting that qqquet's designs might be easier to follow
if he'd describe them more as games and less as mathematical theorems. :-)
So let me try to rephrase the rules for this one.
Pick a size m, make an m-by-m grid. Choose a player to go first. That
player places a 1 in any square of the grid, then in clockwise order
players take turns writing 2, 3, ... in order to m^2, one number per
square.
Then the first player (or maybe just continuing clockwise if m^2 does
not divide evenly by the number of players?) chooses any square and
crosses it out, announcing "1". Continuing clockwise, each player on
his turn crosses out another square, which must be orthogonally
adjacent to at least one square that was already crossed out. Then
he announces any integer divisor of the newly crossed-out number,
*except* the number announced by the immediately preceding player.
He then scores points equal to the absolute difference between his
number and the previous one, and play proceeds to the next player.
When all squares have been crossed out, the LOW score wins.
There, did I miss anything?
Now, as to the game itself: I think you need to ensure an equal number
of turns for each player, unless going early is somehow enough of an
advantage to make up for having to take one more turn than a later
player. But I think there are more serious issues, like the fact that
in a two-player game, one player places all the even numbers in the
first stage, while the other player places all but one of the primes.
The primes are of course critical since they have only two possible
factors, so you either want to use them as "1" or else play a careful
dance hopping between adjacent primes to avoid big deltas, so the
locations of the primes seem far more important than the others.
But that's another thing, how much is really added by doing all this on
a square grid and constraining which squares can be chosen during the
second stage? It affects the game for a while, but eventually more and
more squares are adjacent to the expanding used region. Maybe that's
the idea, and the players jockey for good moves in the early stage and
then mostly alternate between 2 and 1 in the latter part of the game.
But it feels like extra complexity for not a lot of gain. And the whole
first part, filling in the grid, is a lot of effort for little return,
since only a small part of the grid will be used during the early part
of the game -- and that part is chosen by the first player -- with the
rest mostly being available all at once.
How about this as a more streamlined approach:
Write the numbers from 1 to N, where N is a multiple of the number of
players. Pick a starting player, then players take turns choosing numbers
by marking them (with initials or a color), one player per number, until
all the numbers have been chosen. The player who chose "1" then begins
the second phase by crossing off the "1", announcing "1", and scoring 1
point. Play then proceeds *clockwise* (i.e., NOT based on who chose "2"),
with each player in turn crossing off ANY one of his chosen numbers not
yet crossed off, announcing any positive integer divisor of that number
that is NOT equal to the one announced by the previous player, and scoring
the absolute difference between that divisor and the previous one. When
all numbers have been crossed off, low score wins.
Note that this avoids the problem of the "1" being last by requiring it
to be first.
-- Don.
qqquet@xxxxxxxxxxxxxx writes:
Here is a game for any plural number of players. Start with an m-by-m.
grid drawn on paper. (I suggest that m be about 8 to 10 for
beginners.) Draw the grid large enough so that two integers can be
written in each square.
In the first phase of the game, players take turns writing the
positive integers 1 to m^2 in order into the squares of the grid. One
number is placed in any empty square of the grid on each move. (So, if
there are 2 players, one player writes in the odd numbers, and the
other player writes in the even numbers.)
Let the variable d (d for 'divisor') start the second phase of the
game with a value of 1.
At the start of the second phase of the game, player 1 then writes the
value of d, which is 1, alongside any number in the grid (in the same
square as the number).
The players thereafter continue to take turns. On a move, a player
chooses any square of the grid that has not yet had a second number
written in it, but is adjacent to (in the direction of above, below,
right of, or left of) any square that has had a second number written
in it.
He/she then writes down in the square (with one number) any* positive
divisor of the number in that square.
The variable d then becomes that divisor.
* The value of d, however, must change each move. The same divisor
number cannot be written in two squares on two consecutive moves.
The absolute value of the difference between the older recent value of
d (the divisor written by the previous player to move) and the new
value of d (the divisor written by the current player moving) is then
added to the currently moving player's score.
Note: The goal of the game is to get the LOWEST score. So, it is
advantageous to change the value of d by as little as possible on a
move. (Changing the value of d by 1 is the best a player can hope for
on a move.)
The game continues until each square has exactly two numbers in it.
(So, there are a total of m^2 moves in the first phase of the game,
and m^2 moves in the second phase of the game.)
As I said before, the player with the lowest score wins.
I would suggest that the divisor numbers (the values of d) be written
smaller than the numbers written during phase 1 of the game, or be
written in another color than the first numbers placed in the squares.
PS: The only problem I can see with this game is if the last square to
get a second number has a 1 in it, and the previous (next to last)
player to move placed a 1 as the second (divisor) number in some
square. (This is a problem because d must change each move.)
Then, in that case, the second phase of the game ends after m^2 - 1
moves.
Thanks,
Leroy Quet
- References:
- Stepping By Divisors -- Grid Game
- From: qqquet
- Stepping By Divisors -- Grid Game
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