Kangaroo jumps

[Francois]

Being half aussie, I can't resist to not give this one:
you have a Kangaroo on a 2-D plane. It only makes jumps of length pi and 1, and only
along the x-axis and the y-axis (either jump length along the x and y axes).
Assume the plane large enough (larger than pi x pi ), if you pick a random point (x0,y0)
on the plane, and a random small number epsilon, what is the probability that at some stage
P( dist(Kangaroo,(x0,y0)) < epsilon ?

 

[Francois]

OK, looks like you guys don't like Kangaroos. So, I'll reword the problem. Show that if you play with your money and can either win or lose by $1 or by $pi then, 'given enough time', you can hit any amount (positive or negative)

 

[naeron]

hahah Kangaroos are fine dude ... people here enjoying spring vacation..

 

[Francois]

Spring break was 3 weeks ago! ;-)
would be the first time texas was ahead of NYC for anything ;-)

 

[pardasani]

OK, looks like you guys don't like Kangaroos. So, I'll reword the problem. Show that if you play with your money and can either win or lose by $1 or by $pi then, 'given enough time', you can hit any amount (positive or negative)

Okay, let me give it a start atleast...

Lets break the number line into parts :- the set of integers, and the set of non integers... Now its easy to recognize that the problem is a 2 dimensional random walk... For the set of integers, we know that any level can be hit by the "standard" - in the sense that it can go up or down by 1- random walk which is one of the components of the problem --> so we conclude the set of integers can be hit...

If we show that any number = p*1 + q*pi , p and q being integers (I guess there is nothing sacred about pi here, any other irrational would be fine) then we would be through... Just that am not able to show that -- any hints or am I totally off the track...

 

[Francois]

You're right on actually. The idea is to show that pZ + qZ with p/q irrational is dense in R.
So, it works with other numbers than 1 and pi.
You just need to pick a random element x in R, and build a sequence un in pZ+qZ that converges towards x.