## Thursday, September 18, 2008

### Filling In Rectangles

Here is a dumb game (which might help kids learn multiplications and
divisions and about primes, etc.):
2 players.
2 decks of cards, n cards each, the cards in both decks labeled with
the integers from 1 to n (one integer per card).
You also have an m-by-m grid, where m, I suggest, is in the vicinity
of
just over n^(3/2)/2, but m is up to the players.
Players take turns drawing 2 cards, one card from each deck.
Players, after drawing the cards, take the product, k, of the 2 cards'
values.
Players fill in any previously unfilled grid-squares so as to form a
rectangle of area k.
(And the width/height of the rectangle need not be the same as the
cards' numbers, unless 1 and a prime {or another 1} are drawn.)
Players do not have to fill any rectangles if they volutarily skip
their turn or if no rectangle of area k can be filled.
Play continues until either the grid is filled completely or until the
cards have all been drawn, whichever comes first.
3 variations on how to determine winner:
1) Winner has most number of grid-squares filled with their color at
game's end.
2) Winner has most number of rectangles at game's end filled with
their color.
3) Winner is last player to fill in a rectangle.
And, if you want to, maybe you can simply use a standard card-deck
instead.
And you draw 2 cards at a time, assigning appropriate numerical values
to Jack, Queen, King, and Ace, of course.
But how would each winner-determination variation affect strategy?
And what would you suggest for an m based on n?
thanks,
Leroy Quet