You seem to misunderstand the relation between upper bound and estimate. Improving the upper bound by 10% would do nothing to improve the estimate. We'd just fi

You seem to misunderstand the relation between upper bound and estimate. Improving the upper bound by 10% would do nothing to improve the estimate. We'd just find that for instance 5.7% of the sample is legal instead of 5.2%. Generating a 10% bigger sample and filtering out adjacent Kings is way more efficient than complicating the generation process to avoid those instances. The only way to improve the current estimate is to make it more accurate by using a larger sample.