Q. 1 B3.8( 4 Votes )

Mark four numbers forming a square in a calendar:



Add the squares of the diagonal pair and find the difference of these sums:

42 + 122 = 160

112 + 52 = 146

160 – 146 = 14

Explain using algebra, why the difference is 14 always.

Answer :

Let’s use algebra to see this.

Taking the first number in the square as x, the others can be filled as below



the squares of diagonal are


x2 and (x+8)2


(x+7)2 and (x+1)2


Add the squares of the diagonal pair


x2 + (x+8)2


= x2 + (x2+82+2 [Using identity = + + 2]


= 2x2 + 64+16


Add the squares of the other diagonal pair


= (x+1)2+(x+7)2


= (x2 +1 +2 (x2+72+2


= (x2 +1 +2 (x2 +49 +14)


= 2x2 +16+ 50


find the difference of these sums:


= (2x2 + 64+16- (2x2 +16+ 50)


= 2x2 + 64+16- 2x2 -16- 50


= 64-50


= 14


Hence the difference is 14, we can take any number as x; which means this holds in any part of the calendar.


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