## Saturday, September 20, 2008

### Prime-Power Divisors Game

Here is a number game
(which might help some students of basic math learn some of the
prime-power divisors of integers).
Players (2) take turns writing down integers in a list
(called the "play list") as follows:
Player 1 starts by writing down any integer >= 2.
A player on his/her move first makes any list (called
the "divisors list") of prime powers (p^k, k = positive integer),
where the primes being raised to the exponents are not
necessarily distinct,
and where the product of the prime-powers equals the previous
integer in the play list (equals the integer most recently
written down in the list by the player's opponent).
For example, if the last integer in the play list was 24, then
the player currently moving can make the divisors list
2,2,2,3 or 2,4,3 or 8,3.
Next, the player currently moving either adds one or subtracts
one from every prime-power in their divisors list
(the player may add 1 to some prime-powers while subtracting
1 from other prime-powers).
Finally the player multiplies these integers together.
(from the example above) 2,4,3, he/she may now have the product
1*5*2 = 10.
This product must not be 1 and it must not have occurred earlier
in the play list.
So, in the example, 10 must not have occurred earlier in the
play list.
Players continue taking turns until one player cannot move
(because all possible products already exist in the play list).
The last player to move is the game's winner.
Short sample game:
Play list:
4, 5, 6, 2, 3
Player 1 wins.
What is a good strategy for this game?
thanks,
Leroy Quet