How it Works › Forums › Exploration Problems (Interesting Math Problems!) › Exploration Problems Counting Cleverly
This topic contains 6 replies, has 2 voices, and was last updated by Ayush 3 years, 2 months ago.

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Hey Math Circle Students!These math problems are for 20 points and really develop your skills as mathematicians to understand higher order thinking beyond just math competitions and AMC 8/Mathcounts, etc. These focus on observing and proving patterns and making conjectures!
If you don’t understand one of the questions, feel free to post in the questions forum here
1. Find a general formula for the number of rectangles in an $n$by$n$ grid using a clever counting method. First, try it out for small cases like $n=2,3,4,5,6$ and see if you can see a common approach.
2. Do the same for the number of triangles in an $n$by$n$ gride. Prove both of your results.
If you discover a pattern or you manage to prove one of the problems, post your discoveries below! These are really openended so you can post any progress you have made on the problems!
AntonWe can just look at the top left corner of each square inside the n x n grid.
For the 1×1 square, there will be n x n of those. For the 2×2, we can see that the top left corner of the 2×2 square must be in a 4×4 square, so there are 16. As a result, the number of squares in nxn is:
n^2 + (n1)^2 + (n2)^2….(nn)^2, which is just the sum of squares up to n. The formula for this is:
n(n+1)(2n+1)/6Also, how do we use Latex again?
Hey Anton,
Great Start! Unfortunately, you missed an important part of the question that we are trying to count all the rectangles in general, not just the squares so try to fix that formula. For the latex, I have sent you a message regarding how to do it.
 This reply was modified 3 years, 3 months ago by admin.
PATYUSH MAKKAR$x=5$
PATYUSH MAKKARyou can also learn latex at aops(a website)
PATYUSH MAKKARI know how to do the question 1 but what I have got has way to many variables though I have proved my approach in the advanced section
AyushGreat work on the advanced section Pratyush! Refer to my comment there. Also, the proof for the general case surprisingly doesn’t require more than 23 variables, so stay tuned for this solution as well.

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