Keep Squaring

Probability Level 5

If the figure in the left is named figure 1, the one in the middle is named figure 2 and the one in the right is named figure 3, then find the number of quadrilaterals in the 2015 th {2015}^{\text{th}} figure that follows this pattern.


The answer is 8126496.

Ashish Menon by Ashish Menon , 20, India

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1 solution

Ashish Menon
Apr 18, 2016

;) This time it is 2015 th {2015}^{\text{th}} figure and not 2016 th {2016}^{\text{th}}

In figure 1, the number of quadrilaterals = 6 6
In figure 2, the number of quadrilaterals = 15 15
In figure 3, the number of quadrilaterals = 28 28
(For clarification):-
In figure 4, the number of quadrilaterals = 45 45
In figure 5, the number of quadrilaterals = 66 66


So, in n th n^{\text{th}} , the number of quadrilaterals = n + 1 + ( 4 × ∑ i = 1 n i ) n+1 + (4× \displaystyle \sum_{i=1}^n i)

So, in 2015 th {2015}^{\text{th}} , the number of quadrilaterals = 2015 + 1 + ( 4 × ∑ i = 1 2015 i ) 2015+1 + (4× \displaystyle \sum_{i=1}^{2015} i)
= 8126496 quadrilaterals. □ _\square

1 Helpful 0 Interesting 0 Brilliant 0 Confused

There are 8 quadrilaterals that you forgot in figure 1, and they all look like the following:

That brings the total up to 6 + 8 = 14 6 + 8 = 14 . Likewise, your other numbers are too small.

Jon Haussmann - 2 years ago

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