
To find: The total number of squares on the board

Answer to Problem 3P
The total possible squares on the board is 204
Explanation of Solution
Given:
Calculation:
Consider squares by their sizes, with one black/white box having side of unit side.
Squares with side of 8 units:
As 8 is maximum possible side, there is only one possibility
Possible squares
Squares with side of 7 units:
As one
Possible squares
Squares with side of 6 units:
Now, squares with 6 -unit side can have either 2 1-unit edge strip left on one side or both sides.
Such possibility of leaving edges arises on 2 fronts.
Since these are 2 independent outcomes
Thus, the total possible ways
Thus, a pattern can be seen that the number of squares (of a particular size) is square of
So, possible squares with side of 5 units
And, possible squares with side of 4 units
And, possible squares with side of 3 units
And, possible squares with side of 2 units
And, possible squares with side of 1 unit
Last one can be verified by intuition.
Since there are total of 64 small boxes with black or white color, there can be only 64 squares ofside unit 1.
Total number of possible squares is sum of all possible ways above calculates.
Thus, the total number of possible squares is
Hence, the total possible squares on the board is 204
Conclusion:
Hence, the total possible squares on the board is 204
Chapter 0 Solutions
Pre-Algebra Student Edition
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