How many rectangles or triangles.

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











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enter image description here



I have come across numerous questions where I am given the picture such as the above one been asked "how many rectangles are there?". I have even come across some slightly different images that instead of rectangles you are supposed to find the number of triangles. Well, I was thinking whether there is any formula or strategy that is used to solve these problems without having to manually count every shape.



Help would be appreciated.



Thank you :)




combinatorics discrete-mathematics recreational-mathematics




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  • see: math.stackexchange.com/questions/429842/…
    – Emilio Novati
    Apr 18 '15 at 11:53










  • See also: Analysis of how-many-squares and rectangles are are there on a chess board?
    – Martin Sleziak
    Aug 28 at 8:19














up vote
7
down vote

favorite












enter image description here



I have come across numerous questions where I am given the picture such as the above one been asked "how many rectangles are there?". I have even come across some slightly different images that instead of rectangles you are supposed to find the number of triangles. Well, I was thinking whether there is any formula or strategy that is used to solve these problems without having to manually count every shape.



Help would be appreciated.



Thank you :)




combinatorics discrete-mathematics recreational-mathematics




share|cite|improve this question






















  • see: math.stackexchange.com/questions/429842/…
    – Emilio Novati
    Apr 18 '15 at 11:53










  • See also: Analysis of how-many-squares and rectangles are are there on a chess board?
    – Martin Sleziak
    Aug 28 at 8:19












up vote
7
down vote

favorite









up vote
7
down vote

favorite











enter image description here



I have come across numerous questions where I am given the picture such as the above one been asked "how many rectangles are there?". I have even come across some slightly different images that instead of rectangles you are supposed to find the number of triangles. Well, I was thinking whether there is any formula or strategy that is used to solve these problems without having to manually count every shape.



Help would be appreciated.



Thank you :)




combinatorics discrete-mathematics recreational-mathematics




share|cite|improve this question














enter image description here



I have come across numerous questions where I am given the picture such as the above one been asked "how many rectangles are there?". I have even come across some slightly different images that instead of rectangles you are supposed to find the number of triangles. Well, I was thinking whether there is any formula or strategy that is used to solve these problems without having to manually count every shape.



Help would be appreciated.



Thank you :)





combinatorics discrete-mathematics recreational-mathematics





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 29 at 9:39









user1729

16.9k64082




16.9k64082










asked Apr 18 '15 at 11:46









anonymous

5692724




5692724











  • see: math.stackexchange.com/questions/429842/…
    – Emilio Novati
    Apr 18 '15 at 11:53










  • See also: Analysis of how-many-squares and rectangles are are there on a chess board?
    – Martin Sleziak
    Aug 28 at 8:19
















  • see: math.stackexchange.com/questions/429842/…
    – Emilio Novati
    Apr 18 '15 at 11:53










  • See also: Analysis of how-many-squares and rectangles are are there on a chess board?
    – Martin Sleziak
    Aug 28 at 8:19















see: math.stackexchange.com/questions/429842/…
– Emilio Novati
Apr 18 '15 at 11:53




see: math.stackexchange.com/questions/429842/…
– Emilio Novati
Apr 18 '15 at 11:53












See also: Analysis of how-many-squares and rectangles are are there on a chess board?
– Martin Sleziak
Aug 28 at 8:19




See also: Analysis of how-many-squares and rectangles are are there on a chess board?
– Martin Sleziak
Aug 28 at 8:19










1 Answer
1






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up vote
4
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accepted










To have a rectangle, you need 2 horizontal lines and 2 vertical lines. So for your given picture, there are $5choose 2$ choices for two vertical lines. Also $4choose 2$ choices for horizontal lines. So there are $5 choose 2times4choose2$ rectangles in total.



The strategy is to find a way to categorize the things you want to count. Various problems will require various tricks, but you can gain experience by trying to solve them by your own.






share|cite|improve this answer




















  • What does the 5 over the 2 in brackets and 4 over the 2 in brackets mean. Is it just another way to show that it is a combination?
    – anonymous
    Apr 18 '15 at 12:06










  • @anonymous: It is called the binomial coefficient, which you can find at Wikipedia.
    – user21820
    Apr 18 '15 at 12:08










  • So basically, what you're saying is that we can choose 2 of 5 possible x-coordinates 10 ways, and we can choose 2 from the 4 y-coordinates in 6 ways. Thus, this gives a total of 10×6=60 rectangles. Is that right?
    – anonymous
    Apr 18 '15 at 12:55










  • Right. That's what I meant.
    – aNumosh
    Apr 18 '15 at 18:34






  • 1




    In this case we are considering squares as rectangles aren't we?
    – swarm
    Apr 15 '16 at 5:08










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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
4
down vote



accepted










To have a rectangle, you need 2 horizontal lines and 2 vertical lines. So for your given picture, there are $5choose 2$ choices for two vertical lines. Also $4choose 2$ choices for horizontal lines. So there are $5 choose 2times4choose2$ rectangles in total.



The strategy is to find a way to categorize the things you want to count. Various problems will require various tricks, but you can gain experience by trying to solve them by your own.






share|cite|improve this answer




















  • What does the 5 over the 2 in brackets and 4 over the 2 in brackets mean. Is it just another way to show that it is a combination?
    – anonymous
    Apr 18 '15 at 12:06










  • @anonymous: It is called the binomial coefficient, which you can find at Wikipedia.
    – user21820
    Apr 18 '15 at 12:08










  • So basically, what you're saying is that we can choose 2 of 5 possible x-coordinates 10 ways, and we can choose 2 from the 4 y-coordinates in 6 ways. Thus, this gives a total of 10×6=60 rectangles. Is that right?
    – anonymous
    Apr 18 '15 at 12:55










  • Right. That's what I meant.
    – aNumosh
    Apr 18 '15 at 18:34






  • 1




    In this case we are considering squares as rectangles aren't we?
    – swarm
    Apr 15 '16 at 5:08














up vote
4
down vote



accepted










To have a rectangle, you need 2 horizontal lines and 2 vertical lines. So for your given picture, there are $5choose 2$ choices for two vertical lines. Also $4choose 2$ choices for horizontal lines. So there are $5 choose 2times4choose2$ rectangles in total.



The strategy is to find a way to categorize the things you want to count. Various problems will require various tricks, but you can gain experience by trying to solve them by your own.






share|cite|improve this answer




















  • What does the 5 over the 2 in brackets and 4 over the 2 in brackets mean. Is it just another way to show that it is a combination?
    – anonymous
    Apr 18 '15 at 12:06










  • @anonymous: It is called the binomial coefficient, which you can find at Wikipedia.
    – user21820
    Apr 18 '15 at 12:08










  • So basically, what you're saying is that we can choose 2 of 5 possible x-coordinates 10 ways, and we can choose 2 from the 4 y-coordinates in 6 ways. Thus, this gives a total of 10×6=60 rectangles. Is that right?
    – anonymous
    Apr 18 '15 at 12:55










  • Right. That's what I meant.
    – aNumosh
    Apr 18 '15 at 18:34






  • 1




    In this case we are considering squares as rectangles aren't we?
    – swarm
    Apr 15 '16 at 5:08












up vote
4
down vote



accepted







up vote
4
down vote



accepted






To have a rectangle, you need 2 horizontal lines and 2 vertical lines. So for your given picture, there are $5choose 2$ choices for two vertical lines. Also $4choose 2$ choices for horizontal lines. So there are $5 choose 2times4choose2$ rectangles in total.



The strategy is to find a way to categorize the things you want to count. Various problems will require various tricks, but you can gain experience by trying to solve them by your own.






share|cite|improve this answer












To have a rectangle, you need 2 horizontal lines and 2 vertical lines. So for your given picture, there are $5choose 2$ choices for two vertical lines. Also $4choose 2$ choices for horizontal lines. So there are $5 choose 2times4choose2$ rectangles in total.



The strategy is to find a way to categorize the things you want to count. Various problems will require various tricks, but you can gain experience by trying to solve them by your own.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Apr 18 '15 at 11:59









aNumosh

61846




61846











  • What does the 5 over the 2 in brackets and 4 over the 2 in brackets mean. Is it just another way to show that it is a combination?
    – anonymous
    Apr 18 '15 at 12:06










  • @anonymous: It is called the binomial coefficient, which you can find at Wikipedia.
    – user21820
    Apr 18 '15 at 12:08










  • So basically, what you're saying is that we can choose 2 of 5 possible x-coordinates 10 ways, and we can choose 2 from the 4 y-coordinates in 6 ways. Thus, this gives a total of 10×6=60 rectangles. Is that right?
    – anonymous
    Apr 18 '15 at 12:55










  • Right. That's what I meant.
    – aNumosh
    Apr 18 '15 at 18:34






  • 1




    In this case we are considering squares as rectangles aren't we?
    – swarm
    Apr 15 '16 at 5:08
















  • What does the 5 over the 2 in brackets and 4 over the 2 in brackets mean. Is it just another way to show that it is a combination?
    – anonymous
    Apr 18 '15 at 12:06










  • @anonymous: It is called the binomial coefficient, which you can find at Wikipedia.
    – user21820
    Apr 18 '15 at 12:08










  • So basically, what you're saying is that we can choose 2 of 5 possible x-coordinates 10 ways, and we can choose 2 from the 4 y-coordinates in 6 ways. Thus, this gives a total of 10×6=60 rectangles. Is that right?
    – anonymous
    Apr 18 '15 at 12:55










  • Right. That's what I meant.
    – aNumosh
    Apr 18 '15 at 18:34






  • 1




    In this case we are considering squares as rectangles aren't we?
    – swarm
    Apr 15 '16 at 5:08















What does the 5 over the 2 in brackets and 4 over the 2 in brackets mean. Is it just another way to show that it is a combination?
– anonymous
Apr 18 '15 at 12:06




What does the 5 over the 2 in brackets and 4 over the 2 in brackets mean. Is it just another way to show that it is a combination?
– anonymous
Apr 18 '15 at 12:06












@anonymous: It is called the binomial coefficient, which you can find at Wikipedia.
– user21820
Apr 18 '15 at 12:08




@anonymous: It is called the binomial coefficient, which you can find at Wikipedia.
– user21820
Apr 18 '15 at 12:08












So basically, what you're saying is that we can choose 2 of 5 possible x-coordinates 10 ways, and we can choose 2 from the 4 y-coordinates in 6 ways. Thus, this gives a total of 10×6=60 rectangles. Is that right?
– anonymous
Apr 18 '15 at 12:55




So basically, what you're saying is that we can choose 2 of 5 possible x-coordinates 10 ways, and we can choose 2 from the 4 y-coordinates in 6 ways. Thus, this gives a total of 10×6=60 rectangles. Is that right?
– anonymous
Apr 18 '15 at 12:55












Right. That's what I meant.
– aNumosh
Apr 18 '15 at 18:34




Right. That's what I meant.
– aNumosh
Apr 18 '15 at 18:34




1




1




In this case we are considering squares as rectangles aren't we?
– swarm
Apr 15 '16 at 5:08




In this case we are considering squares as rectangles aren't we?
– swarm
Apr 15 '16 at 5:08

















 

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