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October 20, 2016 at 2:07 pm #384
MathJax TeX Test Page
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 open-ended so you can post any progress you have made on the problems!
October 20, 2016 at 6:28 pm #387
We 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 + (n-1)^2 + (n-2)^2….(n-n)^2, which is just the sum of squares up to n. The formula for this is:
Also, how do we use Latex again?October 20, 2016 at 8:17 pm #389
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.
October 23, 2016 at 7:54 am #401
- This reply was modified 1 year, 6 months ago by admin.
$x=5$October 23, 2016 at 7:55 am #402
you can also learn latex at aops(a website)October 23, 2016 at 8:02 am #403
I 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 sectionOctober 25, 2016 at 6:36 am #407
Great work on the advanced section Pratyush! Refer to my comment there. Also, the proof for the general case surprisingly doesn’t require more than 2-3 variables, so stay tuned for this solution as well.November 5, 2017 at 11:59 pm #965
I know in last week’s class we counted the pairs of rows and columns that made the rectangle up. Can we do that for part (a)?