The current pandemic disease coronavirus (COVID-19) has not only become a worldwide health emergency, but also devoured the global economy. Despite appreciable research, identification of targeted populations for testing and tracking the spread of COVID-19 at a larger scale is an intimidating challenge. There is a need to quickly identify the infected individual or community to check the spread. The diagnostic testing done at large-scale for individuals has limitations as it cannot provide information at a swift pace in large populations, which is pivotal to contain the spread at the early stage of its breakouts. Recently, scientists are exploring the presence of SARS-CoV-2 RNA in the faeces discharged in municipal wastewater. Wastewater sampling could be a potential tool to expedite the early identification of infected communities by detecting the biomarkers from the virus. However, it needs a targeted approach to choose optimized locations for wastewater sampling. The present study proposes a novel fuzzy based Bayesian model to identify targeted populations and optimized locations with a maximum probability of detecting SARS-CoV-2 RNA in wastewater networks. Consequently, real time monitoring of SARS-CoV-2 RNA in wastewater using autosamplers or biosensors could be deployed efficiently. Fourteen criteria such as population density, patients with comorbidity, quarantine and hospital facilities, etc. are analysed using the data of 14 lac individuals infected by COVID-19 in the USA. The uniqueness of the proposed model is its ability to deal with the uncertainty associated with the data and decision maker's opinions using fuzzy logic, which is fused with Bayesian approach. The evidence-based virus detection in wastewater not only facilitates focused testing, but also provides potential communities for vaccine distribution. Consequently, governments can reduce lockdown periods, thereby relieving human stress and boosting economic growth.

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Wythoff's Game is a variation of Nim in which players may take an equal number of stones from each pile or make valid Nim moves. W. A. Wythoff proved that the set of P-Positions (losing position), C, for Wythoff's Game is given by C := ( kφ , kφ 2 ), ( kφ 2 , kφ ) : k ∈ Z ≥0 [Wyt07]. An open Wythoff problem remains where players make the valid Nim moves or remove kb stones from each pile, where b is a fixed integer. We denote this as the (b, b) game. For example, regular Wythoff's Game is just the (1, 1) game. In 2009, Duchêne and Gravier [DG09] proved an algorithm to generate the set of P-Positions for the (2, 2) game by exploiting the periodic nature of the differences of stones between the two piles modulo 4. We observe similar cyclic behaviour (see definition 3.2) for any b, where b is a power of 2, modulo b 2 , and construct an algorithm to generate the set of P-Positions for this game. Let a be a power of 2. We prove our algorithm works by first showing that it holds for the first a 2 terms in the (a, a) game. Next, we construct an ordered multiset for the (2a, 2a) game from the a 2 terms, and an inductive proof follows. Moreover, we conjecture that all cyclic games require a to be a power of 2, suggesting that there is no similar structure in the generalised (b, b) game where b isn't a power of 2. Future directions for generalising this result would likely utilise numeration systems, particularly the PV numbers.