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In-how-many-ways-we-can-choose-3-squares-on-a-chess-board-such-that-one-of-the-squares-has-its-two-sides-common-to-other-two-squares-

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Question Number 22043 by Tinkutara last updated on 10/Oct/17
In how many ways we can choose 3 squares on a chess board such that one of the squares has its two sides common to other two squares?
$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{we}\:\mathrm{can}\:\mathrm{choose}\:\mathrm{3} \\ $$$$\mathrm{squares}\:\mathrm{on}\:\mathrm{a}\:\mathrm{chess}\:\mathrm{board}\:\mathrm{such}\:\mathrm{that}\:\mathrm{one} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{squares}\:\mathrm{has}\:\mathrm{its}\:\mathrm{two}\:\mathrm{sides}\:\mathrm{common} \\ $$$$\mathrm{to}\:\mathrm{other}\:\mathrm{two}\:\mathrm{squares}? \\ $$
Answered by Rasheed.Sindhi last updated on 10/Oct/17
Let the square which shares its two sides is named as ′main′. Case-1:When the main is on the corner. 4 squares Making corner square as main only one triple of squares can be slected. 4 corners→4 triples of squares. Case-2:When the main is on the edge but not the corner.24 squares Making edge square (without corners) as main 3 triples of squares can be acieved. 24 squares→72 triples. Case-3:When the main is other than corner and edge squares. 36 squares. Making any non-edge square as main 6 triples can be achieved. 36 squares→216 triples. 4+72+216=292 triples of squares. In 292 ways such triples of squares can be chosen.
$$\mathrm{Let}\:\mathrm{the}\:\mathrm{square}\:\mathrm{which}\:\mathrm{shares}\:\mathrm{its}\: \\ $$$$\mathrm{two}\:\mathrm{sides}\:\mathrm{is}\:\mathrm{named}\:\mathrm{as}\:'\mathrm{main}'. \\ $$$$\mathrm{Case}-\mathrm{1}:\mathrm{When}\:\mathrm{the}\:\mathrm{main}\:\mathrm{is}\:\mathrm{on}\: \\ $$$$\mathrm{the}\:\mathrm{corner}.\:\mathrm{4}\:\mathrm{squares} \\ $$$$\:\mathrm{Making}\:\mathrm{corner}\:\mathrm{square}\:\mathrm{as}\:\mathrm{main}\: \\ $$$$\mathrm{only}\:\mathrm{one}\:\mathrm{triple}\:\mathrm{of}\:\mathrm{squares}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{slected}. \\ $$$$\:\mathrm{4}\:\mathrm{corners}\rightarrow\mathrm{4}\:\mathrm{triples}\:\mathrm{of}\:\mathrm{squares}. \\ $$$$\mathrm{Case}-\mathrm{2}:\mathrm{When}\:\mathrm{the}\:\mathrm{main}\:\mathrm{is}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{edge}\:\mathrm{but}\:\mathrm{not}\:\mathrm{the}\:\mathrm{corner}.\mathrm{24}\:\mathrm{squares} \\ $$$$\mathrm{Making}\:\mathrm{edge}\:\mathrm{square}\:\left(\mathrm{without}\right. \\ $$$$\left.\mathrm{corners}\right)\:\mathrm{as}\:\mathrm{main}\:\mathrm{3}\:\mathrm{triples}\:\mathrm{of} \\ $$$$\mathrm{squares}\:\mathrm{can}\:\mathrm{be}\:\mathrm{acieved}. \\ $$$$\mathrm{24}\:\mathrm{squares}\rightarrow\mathrm{72}\:\mathrm{triples}. \\ $$$$\mathrm{Case}-\mathrm{3}:\mathrm{When}\:\mathrm{the}\:\mathrm{main}\:\mathrm{is}\:\mathrm{other} \\ $$$$\mathrm{than}\:\mathrm{corner}\:\mathrm{and}\:\mathrm{edge}\:\mathrm{squares}. \\ $$$$\mathrm{36}\:\mathrm{squares}. \\ $$$$\mathrm{Making}\:\mathrm{any}\:\mathrm{non}-\mathrm{edge}\:\mathrm{square} \\ $$$$\mathrm{as}\:\mathrm{main}\:\mathrm{6}\:\mathrm{triples}\:\mathrm{can}\:\mathrm{be}\:\mathrm{achieved}. \\ $$$$\mathrm{36}\:\mathrm{squares}\rightarrow\mathrm{216}\:\mathrm{triples}. \\ $$$$\mathrm{4}+\mathrm{72}+\mathrm{216}=\mathrm{292}\:\mathrm{triples}\:\mathrm{of}\:\mathrm{squares}. \\ $$$$ \\ $$$$\mathrm{In}\:\mathrm{292}\:\mathrm{ways}\:\mathrm{such}\:\mathrm{triples}\:\mathrm{of}\:\mathrm{squares} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{chosen}. \\ $$
Commented by Tinkutara last updated on 11/Oct/17
Thank you very much Sir!
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

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