Take nine numbers forming a square in a calendar and mark the four numbers at the corners.
Multiply the diagonal pairs and find the difference of these products.
3 × 19 = 57
17 × 5 = 85
85 – 57 = 28
Explain using algebra, why the difference is always 28 (It is convenient to take the number at the centre as x).
Let’s use algebra to see this.
Taking the first number in the square as x, the others can be filled as below
Multiply the diagonal pairs
(x)(x+16)
= x2+16x
Other diagonal product
(x+14)(x+2) [using identity (x+y)(u+v)= xu+xv+yu+yv]
= x2+2x+14x+28
= x2+16x+28
Difference of these products
=(x2+16x+28) - (x2+16x)
=x2+16x+28 - x2-16x
= 28
Hence the difference is 28, we can take any number as x; which means this hold in any part of the calendar.
We can take x at the center, but this will complicate our calculations.
Couldn't generate an explanation.
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