Concentric Circles Game (multiply and divide)

I made this simple game up the other day,
in part so as to help teach kids/adults basic math concepts,
but it might be far more useful as useless entertainment...
if even that...
(You might want to share this game with the kiddies you know,
after explaining the rules to them in more kid-friendly words...
I know not about age-appropriateness.
But give the younger kids some credit for being able to understand
this
game,
at least if explained properly.. :)
{But they should probably have some basic idea of multiplication
and of *simple* division, at least}.)
(Note: uncirclular ascii-art image below to help with visualization.)
Start with m concentric circles drawn on paper (looking like a
target),
where m is an integer from 5 to 12 (or higher perhaps).
The rings between the circles are subdivided so that the inner ring
(which is just a circle/disk) is subdivided into 2 equal-area
sections,
the next ring out is subdivided into 3 equal sections,
the next ring into 4, etc..
until the outer ring is subdivided into (m+1) equal sections.
(And the sections are aligned so that one boundry of each is along a
single line.)
(For serious mathematical people:
By "ring", I of course mean a GEOMETRIC ring, ie. an annulus.)
Players start by each placing a game-piece (small enough to fit in the
smallest section) into any unoccupied section on the board.
A spinner or dice are used to pick a random integer n, where each n
should be between 1 and somewhere near m.
On each move, a player can place a new game-piece (leaving earlier
pieces) k rings inward (if possible) from their last game-piece's
position, or k rings outwards (if possible),
where k is any integer which divides into n evenly.
("Outward" and "inward" are along a chain of sections which touch. Do
not pass center of circle or outer circle.)
(So, if n = 4, for example, you can move 1, 2, or 4
{if possible} rings outward or inward.)
OR the player can place a game-piece clockwise or counterclockwise (in
the same ring)
a *multiple* of n spaces from their last-placed piece.
(With n=4, for example, the player can place his next piece
4, 8, 12, 16,...positions clockwise or counterclockwise of
her/his last-placed piece.
And, no, this does *not* mean that every section in that ring
is *necessarily* reachable in that move!)
So, again, on each of her/his moves, a player can choose, after n is
picked, to either:
Move counterclockwise or clockwise any *multiple* of n segments;
OR
Move outwardly or inwardly any *divisor* of n rings.
But in any case, a player can't change directions during his/her move.
(And the player's {counter}clockwise moves may indeed circle around a
single ring as many times as needed.)
A player can "jump over" occupied spaces when measuring the number of
spaces from their last piece, but the player must ONLY place a piece
on an
unoccupied section.
And the winner is the last player able to place a piece on the board.
Finally, I will give an example of where a player can move,
using this rectified gameboard:
(Use fixed-width font.
Imagine board curved, and left and right sides joined so game-board is
circle.
Bottom is center.)
-----------------------------------
! * ! * ! o ! @ ! @ ! * ! o ! ... !
------------------------------------
!.. !. * .!. @ .! . @.!. .!. .! .. !
------------------------------------
! .@.! .. ! ..x..!..*.. !.@. !. o. !
------------------------------------
! ..*..! ..@.. ! ..@.. !..o..!....!
------------------------------------
!...@ ...! ...@...!...*.. ! ..... !
------------------------------------
!....*.....!..........!.....o......!
------------------------------------
!......@ ....... ! .......o....... !
....................................
Game is player-* vs player-o (or sometimes x, sometimes @).
It is o's move. His last piece is the 'x'.
He has just rolled a 4 .
So, I have denoted the sections he can put his next piece at by '@'s,
which are now all currently unoccupied sections.
thanks,
Leroy Quet