perplexus.info :: Games : Box game

Box game (Posted on 2005-01-04)

My friend and I used to play a simple game. An abitrarily large array of dots was drawn on paper, and we took turns connecting adjacent dots vertically or horizontally. Whenever a box connecting four adjacent dots was made, the player who finished it got an extra turn and a point. When all possible lines were drawn, the game ended and the one with the most points won.

My friend and I were both horrible at this game; we both used the same ineffective strategy. On each of our turns, when possible, we would always make a move that would not allow the other player to make a box the next turn.

Using this strategy and 25 dots in a 5x5 grid, what is the fewest number of moves possible before someone has to let the other player score? What if we use 36 dots in a 6x6 grid? And 49 dots in a 7x7 grid?

Submitted by Tristan

Rating: 2.6667 (6 votes)

Solution: (Hide)

15 moves 23 moves 31 moves _|_ _|_ _| | | |_ _| |_ _| |_ _ | _ |_ _ _| _ _ _ _ _ | _ _| |_ |_ | _| | | | _ | | _ _| | |_ | | | | _ _|_ _ | | | |
Method:
At first glance, it seems that each box must have two of the sides drawn in. If we make each line touch two boxes,then we should be able to reach the "saturation" point (where no more lines can be added) in (n-1)^2 moves, where n is the number of dots per side.
However, in the above solutions, we can see that there are some boxes with only 1 side drawn in or none at all. And yet, no more sides can be drawn in. To reach the saturation point fastest, we must maximize the number of boxes like this. I believe the above solutions are the best possible.
For further methods and thoughts, read the comments.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
No Subject	CodyDunning	2024-07-19 07:18:58
Puzzle Thoughts	K Sengupta	2023-08-04 13:34:23
7 x 7 improvement	Hugo	2005-03-24 17:09:42
re: I surrender	Tristan	2005-03-23 00:50:51
I surrender	Hugo	2005-03-22 17:02:11
this many?	Brandon	2005-03-22 15:33:57
re(3): ...and going...	Hugo	2005-02-09 16:14:50
re(2): ...and going...	David Shin	2005-02-09 15:13:20
re: ...and going...	Hugo	2005-02-09 11:08:59
...and going...	Tristan	2005-02-09 00:00:56
re: Proof of optimality of 15 for 5x5	Hugo	2005-01-24 18:56:13
Proof of optimality of 15 for 5x5	David Shin	2005-01-15 18:24:57
Thoughts on the maximum	David Shin	2005-01-15 17:43:33
Towards a Grand Unifying Formula, HELP WANTED	Hugo	2005-01-14 18:42:22
7 x 7 in 32 moves	Hugo	2005-01-13 13:19:15
7 x 7 in 34 moves	Hugo	2005-01-13 10:02:26
7 x 7 in 35 moves	Hugo	2005-01-13 09:24:31
5 x 5 in 15 moves	Hugo	2005-01-12 13:35:25
Keep on going	Tristan	2005-01-11 05:07:51
re: Solution 6x6 grid	Milind	2005-01-06 07:30:35
A probable solution	Milind	2005-01-06 07:16:38
Solution 6x6 grid	Hugo	2005-01-05 19:35:37
No Subject	Jer	2005-01-05 18:57:12
%$! Aahrg FEWEST!	Hugo	2005-01-05 09:06:27
re: In conclusion	Steve Herman	2005-01-05 04:59:12
How to draw in a comment	Tristan	2005-01-05 01:42:57
A goal to aim for	Tristan	2005-01-05 01:18:17
In conclusion	Gamer	2005-01-04 21:10:15
Solution	Milind	2005-01-04 20:41:01
Some more moves	Hugo	2005-01-04 20:25:18
I will attempt a picture	Gamer	2005-01-04 20:20:03
Efficiency	Gamer	2005-01-04 20:19:30
Possible Solution	Eric	2005-01-04 19:27:45

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