Wednesday, July 22, 2009

Sequence Ascension Solitaire Grid Game

This is a game/challenge for one player. Start by drawing an n-by-n grid on paper. (n could be about 5 to 10, maybe.)

Next, either use a pre-existing integer sequence (such as {a(k)}, where a(n) = the number of divisors of n), or randomly pick the integers in the sequence.
If randomly picking the sequence, first write down n^2 numbers in a list before continuing to the next part of the game.
It might help to first write down the numbers even if you are using a preexisting sequence. Write down n^2 numbers, just in case they are needed.

Next, place the integers into the grid, starting at any square, and then placing each number -- in order by the order of the indexes (a(1), then a(2), then a(3)...etc), each number in any empty square horizontally, vertically, or diagonally adjacent to the last square filled with a number.
(a(k) is always next to a(k-1).)
Continue doing this until you cannot place any more integers (because there are no empty squares next to where you last put a number).

Next, starting at any square with a number in it, draw a path of connected line segments from square to adjacent square -- adjacent and in the direction of either vertical, horizontal, or diagonal -- such that each number drawn to is greater than or equal to the number in the previous square of the path.
(The numbers of the path never descend.)
The path must not visit any square more than once. But two diagonal segments of the path may cross.

Move until you can't move anymore. (The last square visited by the path will not be bordered by any unvisited square with a number >= the value in the last square.)

Your score is the number of squares your path visits.

Note: I realize that you could use the all-1's sequence, say, and score a perfect n^2 points each time, but that wouldn't be much fun.

As an easy challenge to myself I used the first 16 terms of the number-of-divisors sequence (1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5)
and a 4-by-4 grid. I got a top score of 13 (in several ways).
Can you do better?

Leroy Quet

Dots To Primes Game

This is a game for any plural number of players.

Materials: Blank pieces of paper, and a grid drawn on tracing paper. The horizontal rows of the grid are labeled in order 1, 2, 3, 4,..., such that there is one number per row.

Players take turns being the offense player.
At the beginning of a round, all of the players take turns placing dots on a blank piece of paper anywhere (anywhere where there isn't already a dot) within a large circle drawn on the paper. The size of the circle is the same each round.
A fixed total number of dots are drawn. This number is the same for all rounds.

After the dots are drawn, the offense player then rotates the tracing-paper grid in any way he/she desires, and places it over the circle of dots such that the circle is completely covered by the grid.

The offense player then reads the vertical positions of the dots from left to right -- relative to the grid. The offense player then reads the vertical positions of the dots from left to right -- relative to the grid. The offense player the forms the "first list" by writing down the numbers of the rows the dots fall into in order from the leftmost dot, relative to the grid, to the rightmost dot.

The offense player then forms a second list of partial sums of the first list.
The offense player starts this second list of numbers by first writing down the first number of the first list of numbers. He/she then adds the next number of the first list to the first number of the second (and of the first) list, and writes down the sum, then continues writing down all the partial sums, summed from left to right, until, finally, the last number in the second list is the sum of all the numbers in the first list.

Then the offense player circles all of the primes in the second list (the list of partial sums). The number of primes is the offense player's score for the round.

Then there is a new round with another offense player. Play continues until each player has been offense the same predetermined number of rounds.

Highest score wins.

Leroy Quet