This Is Geometry, Isn't It?

Probability Level 3

If the number of quadrilaterals in an n × n n\times n regular grid is x x , and the number of quadrilaterals in an n × n n\times n slope grid is y y , then what can you conclude about x x and y y ?

Clarification:

  • A quadrilateral is a polygon that has 4 sides.
This is one part of Quadrilatorics .
x < 2 y x<2y x > 2 y x>2y x = 2 y x=2y Pancake Not enough information

Kenneth Tan by Kenneth Tan , 21, Malaysia

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1 solution

Laurent Shorts
Apr 4, 2016

x = n 2 ( n + 1 ) 2 4 x=\frac{n^2(n+1)^2}{4} and y = ( n − 1 ) n ( n + 1 ) ( n + 2 ) 24 r e c t a n g l e s + ( n − 1 ) n ( n + 1 ) 3 t r a p e z i u m s = ( n − 1 ) n ( n + 1 ) ( n + 10 ) 24 y=\frac{(n-1)n(n+1)(n+2)}{24}\mathrm{ rectangles }+\frac{(n-1)n(n+1)}{3}\mathrm{ trapeziums }=\frac{(n-1)n(n+1)(n+10)}{24} .

3 Helpful 0 Interesting 0 Brilliant 0 Confused

y y is ( n − 1 ) n ( n + 1 ) ( n + 10 ) 24 \frac{(n-1)n(n+1)(n+10)}{24} , not ( n − 1 ) n ( n + 1 ) ( n + 2 ) 24 \frac{(n-1)n(n+1)(n+2)}{24} .

Kenneth Tan - 4 years, 4 months ago

You're right, I counted only the rectangles… There's a ( n − 1 ) n ( n + 1 ) 3 \dfrac{(n-1)n(n+1)}{3} number of trapeziums to add to it.

Laurent Shorts - 4 years, 4 months ago

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