EDIT: Seems the end result under solely provides an higher sure (though fairly an honest one), and it is not truly potential to achieve with precise strikes given within the query. Leaving the reply up in any case, because the strategy is perhaps helpful for others. Because of @TimSeifert for pointing this out within the feedback.
When you place the preliminary cell on the origin, after which for each coordinate level (x, y) you calculate their Manhattan distance d = |x| + |y| from the origin and assign a multiplier of $ frac{1}{2^d} $ to every level,
like so
1/8
1/8 1/4 1/8
1/8 1/4 1/2 1/4 1/8
1/8 1/4 1/2 1 1/2 1/4 1/8
1/8 1/4 1/2 1/4 1/8
1/8 1/4 1/8
1/8
then, at each transfer, the overall quantity of cell on the grid will keep the identical, or enhance for those who transfer suboptimally.
This property seems to be very helpful, as a result of if we now add up all of the multipliers alongside the x axis we get 1/2^n + … 1/4 + 1/2 + 1 + 1/2 + 1/4 … + 1/2^n which approaches 3 when n approaches infinity, and equally including up all of the horizontal traces (y = integer fixed), we get the identical end result multiplied by 3, for a sum of
9
models of multiplier over your complete x/y airplane.
The unique cell beginning on the origin will all the time take (at the least) one unit of it, so we’re left with a potential most of
8 models of empty circle.
That seems to be a surprisingly small circle. Sorting all of the coordinate factors by their (cartesian) distance c = $sqrt{x^2+y^2}$ from the origin, and tallying up their multipliers, we get
distance c | variety of factors | sum of multipliers | cumulative sum 0 | 1 | 1 | 1 1 | 4 | 2 | 3 sqrt(2) | 4 | 1 | 4 2 | 4 | 1 | 5 sqrt(5) | 8 | 1 | 6 2*sqrt(2) | 4 | .25 | 6.25 3 | 4 | .5 | 6.75 sqrt(10) | 8 | .5 | 7.25 sqrt(13) | 8 | .25 | 7.5 4 | 4 | .25 | 7.75 sqrt(17) | 8 | .25 | 8.00
Which means with optimum play, all of the factors at distance 4 or much less will be empty, and there’ll unavoidably be cells at distance
$mathbf{sqrt{17}}$, as a result of we are able to solely make arbitrarily many (not infinitely many) strikes,
which is thus the radius of the maximal empty circle.
(Disclaimer: all of this after all hinges on the unlikely assumption that I made no errors whereas manually including all that up :-] )
