The Total Numbers Of Square On A Chessboard Is - Vedantu

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AnswerQuestion Answers for Class 12Class 12 BiologyClass 12 ChemistryClass 12 EnglishClass 12 MathsClass 12 PhysicsClass 12 Social ScienceClass 12 Business StudiesClass 12 EconomicsQuestion Answers for Class 11Class 11 EconomicsClass 11 Computer ScienceClass 11 BiologyClass 11 ChemistryClass 11 EnglishClass 11 MathsClass 11 PhysicsClass 11 Social ScienceClass 11 AccountancyClass 11 Business StudiesQuestion Answers for Class 10Class 10 ScienceClass 10 EnglishClass 10 MathsClass 10 Social ScienceClass 10 General KnowledgeQuestion Answers for Class 9Class 9 General KnowledgeClass 9 ScienceClass 9 EnglishClass 9 MathsClass 9 Social ScienceQuestion Answers for Class 8Class 8 ScienceClass 8 EnglishClass 8 MathsClass 8 Social ScienceQuestion Answers for Class 7Class 7 ScienceClass 7 EnglishClass 7 MathsClass 7 Social ScienceQuestion Answers for Class 6Class 6 ScienceClass 6 EnglishClass 6 MathsClass 6 Social ScienceQuestion Answers for Class 5Class 5 ScienceClass 5 EnglishClass 5 MathsClass 5 Social ScienceQuestion Answers for Class 4Class 4 ScienceClass 4 EnglishClass 4 MathsThe total numbers of square on a chessboard is:A) \[206\]B) \[205\]C) \[204\]D) \[202\]AnswerVerified568.5k+ viewsHint: At first, we have to find out how many positions there are that each size of square can be located.A chess board contains \[1 \times 1\], \[2 \times 2\],\[3 \times 3\], \[4 \times 4\],\[5 \times 5\],\[6 \times 6\],\[7 \times 7\],\[8 \times 8\] square located in different places though can only fit in 1 position vertically and 1 horizontally. We can find the locations for those squares, then we can find the sum of squares.Complete step-by-step solution:We have to find the total number of squares on a chessboard.At first, we have to find out how many positions there are that each size of square can be located.For example, a \[1 \times 1\] square can be located in 8 locations horizontally and 8 locations vertically that is in 64 different positions. An \[8 \times 8\] square though can only fit in 1 position vertically and 1 horizontally.For example, a \[2 \times 2\] square can be located in 7 locations horizontally and 7 locations vertically that is in 49 different positions. An \[7 \times 7\] square though can only fit in 2 positions vertically and 2 horizontally.So, we can prepare a table such as:

Size	Horizontal position	Vertical position	Positions
\[1 \times 1\]	8	8	64
\[2 \times 2\]	7	7	49
\[3 \times 3\]	6	6	36
\[4 \times 4\]	5	5	25
\[5 \times 5\]	4	4	16
\[6 \times 6\]	3	3	9
\[7 \times 7\]	2	2	4
\[8 \times 8\]	1	1	1
Total	204

Hence, the total number of squares on a chessboard is 204.Hence, the correct option is C.Note: It is clear from the above analysis that the solution in case of \[n \times n\] is the sum of the squares from \[{n^2}\] to \[{1^2}\] that is to say \[{n^2} + {(n - 1)^2} + {(n - 2)^2} + {(n - 3)^2} + ... + {2^2} + {1^2}\]For a chessboard, \[n = 8\]So, the total number of squares is \[{8^2} + {7^2} + {6^2} + {5^2} + ... + {2^2} + {1^2}\]Solving we get, the total number of squares is \[ = 204\]Recently Updated PagesMaster Class 11 English: Engaging Questions & Answers for SuccessMaster Class 11 Biology: Engaging Questions & Answers for SuccessMaster Class 11 Physics: Engaging Questions & Answers for SuccessMaster Class 11 Maths: Engaging Questions & Answers for SuccessMaster Class 11 Accountancy: Engaging Questions & Answers for SuccessClass 11 Question and Answer - Your Ultimate Solutions GuideMaster Class 11 English: Engaging Questions & Answers for SuccessMaster Class 11 Biology: Engaging Questions & Answers for SuccessMaster Class 11 Physics: Engaging Questions & Answers for SuccessMaster Class 11 Maths: Engaging Questions & Answers for SuccessMaster Class 11 Accountancy: Engaging Questions & Answers for SuccessClass 11 Question and Answer - Your Ultimate Solutions Guide

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