Dear Sam, After the sequence you give has occurred twice a jump would have to be duplicated. Someone else has also suggested that jumps should be unidirectional, say always from left to right. R. On Fri, 26 Jun 2015, Sam Benner wrote:

I think this game should never end.  Consider the following situation: 011011 001111 011110 011011 It's possible that any other moves would be illegal or require playing a bean, and this can happen on any game with 6 or more spaces.  As it is in the interest of one player to set something like this up, it may be that this is the most common ending.  I'm not certain how to find these situations - there needs to be no other options for moves unless they also have similar results.

Potentially, the solution would be to require beans to move in one direction only, as we would satisfy an ending condition then.

Sam

On Fri, Jun 26, 2015 at 3:28 PM, rkg <rkg@ucalgary.ca> wrote: To save duplicated effort, here are two messages from Richard Nowakowski, who has used Aaron Siegel's CG-Suite to analyze Leap Frog as far as n = 11.    R.

---------- Forwarded message ---------- Date: Fri, 26 Jun 2015 19:15:09 +0000 From: Richard Nowakowski <R.Nowakowski@Dal.Ca> To: rkg <rkg@ucalgary.ca> Subject: Re: Leap Frog

Richard,

if I understood the game correctly then CGSuite gives: [0] = *

[0,0] = 0

[0,0,0] = *

[0,0,0,0] = 0

[0,0,0,0,0] = *

[0,0,0,0,0,0] = 0

[0,0,0,0,0,0,0] = *2

[0^8] = 0

[0^9] = *

[0^10] = 0

The options are:  n=1, 0 n=2, all * n=3, all 0 n=4, [1,0,0,0]=[0,0,0,1] =* other two are *2 n=5, all 0 n=6,  all * n=7, all * except [0,0,0,1,0,0,0] = *2 n=8, all * n=9 all *2 except [1,0^8], [0^4,1,0^4], [0^8,1] which are 0 n=10, all *

I'm waiting on n=11 and n=12.

No insights though.

Second message:

The game is `symmetric' so I'll only give half the options.

n=11 is * but [1,0^{10}] = [0,1,0^9] = [0^5,1,0^5] = 0 [0,0,1,0^8] = *2 [0,0,0,1,0^7] = *6 [0^4,1,0^6]=*5

Cheers,

Richard. ________________________________________ From: rkg <rkg@ucalgary.ca> Sent: Friday, June 26, 2015 3:12 PM To: math-fun Subject: Leap Frog

To whom it may concern.

Here is a game, Leap Frog, which may not be original.

It is played on a strip of squares.  A move is to place a bean on an unoccupied square.  So far we have one of the many manifestations of She-Loves-Me-She-Loves-Me-Not, but there is a second type of move.  A player may jump a bean onto an unocupied square, provided there is at least one intervening square and all such intervening squares are occupied by beans.  To ensure that the Ending Condition is satisfied, we will say that such jumping moves may not be reversed.  If you make that ``may not be IMMEDIATELY reversed'' or ``may not be reversed by the opposite player'' then these versions may be of interest, but I suspect that ties occur for n > 5.

For 0 < n < 5 the game is equivalent to S-L-M-S-L-M-N, since jumping moves need not arise.  For example, for n = 4, 1 & 3 are good replies to one another, as are 2 & 4.

n = 5 is an N-position.  For example, placing a bean on square 3 wins. The moves 1 & 5 are good replies to each other, while 2 (or 4) can be met by 1 (or 5), since the jumping moves 2 to 4 or 1 to 4 are  met by the jump 3 to 5.

n = 6 is beginning to get difficult by hand.  Over to you!

Fame & fortune await a complete analysis!  R.