## Thursday, September 18, 2008

### Coprimality: 1-person game

A 1-person game/ puzzle (with a score), which involves a little math.
I suggest a grid of at least n=5, and (for serious fun) perhaps an n
of at least 10.
So, you start by writing 1 in any of the grid's n^2 squares.
You then write the 2 adjacent to the 1, the 3 adjacent to the 2, the 4
adjacent to the 3, etc, in a chain of increasing integers, in such
away that (ideally...) all n^2 integers are placed into the grid, one
integer (and ONLY one integer) per square.
(By "adjacent", I mean immediately next to in the direction of up,
down, left, or right, but *not* diagonally.)
And, oh by the way, the integers are to be placed so that EVERY pair
of adjacent squares contains two integers which are COPRIME with each
other...
(Again, "adjacent" is as defined above.)
So, you place the integers into the grid as far as you can until your
path cannot be continued for whatever reason.
And your score is the last integer you were able to write into a
square.

Here are my best, as of now, n=6 game, for examples:
21 22 25 26 27 *
20 23 24 * 28 29
19 18 1 2 3 *
* 17 6 5 4 *
15 16 7 8 9 *
14 13 12 11 10 *
(My score = 29)
And better,
1 2 3 34 27 26
6 5 4 33 28 25
7 8 9 32 29 24
12 11 10 31 30 23
13 14 17 18 * 22
* 15 16 19 20 21
(My score: 34)

(And I apologize if I missed that any 2 adjacent integers in my
examples have a GCD which is greater than 1.)
(And, double-check every time you write down an integer so as to make
sure that it is coprime with each of its neighbors, I suggest!
{unless playing on a computer, which should be checking for
coprimality for you})

Thanks,
Leroy Quet