The essence of the problem is linked to another problem of the theory numbers!


We have a path from an infinite set of square plates. A traveler wants to go all the way and step on all stoves.
Attempts to pass a traveler, an infinite number of times. It should be with each new attempt to increase the length of the step!


The traveler passes the path length step $\displaystyle F_{1}$
The traveler passes the path with the length of step $\displaystyle F_{2}$.
And so, infinitely more. With each new attempt, Wayfarer increases stride length!



Following the path with the length step $\displaystyle F_{1}$ The traveler walked an average $\displaystyle S_{1}$- 1 plate.

Following the path with the length of step $\displaystyle F_{2}$ The traveler walked an average of $\displaystyle S_{2}$-1,28 plates.

Here's how the process path. (We take this as an axiom.)
Initially, for example, after you pass the path with the length of step $\displaystyle F_{1}$, we have this average between the steps was 1. $\displaystyle S_{1}$
Next. Wayfarer increased stride length $\displaystyle F_{2}$. And if he had not come for a single plate, then the value would increase between steps in $\displaystyle \frac{F_{2}}{F_{1}}$` and would be equal to 1,4. $\displaystyle W_{2}$
But he comes to the new plates, and this value `$\displaystyle W$` is reduced by 0,12 $\displaystyle Y_{2}$ and is equal to 1,28 $\displaystyle S_{2}$


In general, this operation is described as follows:

$\displaystyle S = W - Y$

Assume that we have established the limits of these values:
$\displaystyle S$ - limit plus infinity.
`$\displaystyle W$ - limit plus infinity.
`$\displaystyle Y$ - limit 0.

Question: for how many squares do not come Wayfarer ".


There is an assumption: Attack is always the first number plates, and the traveler comes to all plates. But this assumption. But the math leads us to what is plus infinite value. What to do?
1.Take assumption?
2.Take mathematical proof?
3.Take both options together, but when put in the logic that the desire to plus infinity, eventually leads to 0!