ArcMap Create Random Points not creating enough points with minimum spacing

[Adopkin]

I'm trying to create a maximum number (or near max) of random points in a polygon with a minimum spacing of 2km between them.  I am using the 'Create Random Points' Data Management tool and selecting the layer (1 polygon) I want the points in as the constraining feature class.  The polygon would easily fit about 50 datapoints spaced by 2km each. 

I set the maximum points (number of points - long) to 20, minimum spacing to 2km, and run the command.  However, with 2km spacing it gives me "WARNING: 000983: The specified number of points could not be created in all cases due to restrictions from the minimum allowed distance." and it usually only produces about 12-15 datapoints, even though it could fit 50 and there's clearly space left in the polygon.  

When I lower the minimum distance threshold to 1km and increase the maximum number of points they usually all fit in the polygon.  Is there a way to get this to work at 2km or another tool that will do this?

[DanPatterson]

Expanding the extent or reducing the spacing are the only options for that tool.

Alternately, the most "densely" packed spacing would be points spaced on a rectangular grid pattern, which would effectively remove the randomness from the location.

[Adopkin]

Thanks for the input!  My solution thus far has been to lower the threshold but I am hoping there is another work around.

Are there any other tools that might be able to do this/put points in a little more densely?   It's mostly that there is a significant amount of space left that this tool won't use.  Picture shows 16 points generated out of 20 requested with a lot of remaining space.

Thanks again!

[DanPatterson]

Perhaps just generate a bunch of points with a closer spacing, then selectively remove the ones with the smallest interpoint distance first... kind of defeats the purpose of randomness, hence my original suggests are still the best recommendations.  You have to decide what is more important.

Of course... randomness implies that you may get luck and get your 20 points IF it is all possible given your criteria.  The quickest check of that is to see if a regular pattern of points will even permit 20