Count the number of shapes in a polyhedron.

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So this is a question that was asked in the International Kangaroo Math Contest 2017. The question is:




The faces of the following polyhedron are either triangles or squares. Each triangle is surrounded by $3$ squares and each square is surrounded by $4$ triangles. If there are $6$ square faces, how many triangular faces are there?






What I did:



Each square shares each of its four neighboring triangles with two more squares. So we can say that for 6 squares we have $6times4 - 2 times 6 = 12$ triangles. However, I still know that this calculation of mine is quite wrong and based on an awkward thinking. So, what is the correct answer and how?



Thanks for the attention.






combinatorics geometry 3d surfaces polyhedra







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  • Can you now accept my answer? (click the green tick below the score next to my answer.)
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up vote
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So this is a question that was asked in the International Kangaroo Math Contest 2017. The question is:




The faces of the following polyhedron are either triangles or squares. Each triangle is surrounded by $3$ squares and each square is surrounded by $4$ triangles. If there are $6$ square faces, how many triangular faces are there?






What I did:



Each square shares each of its four neighboring triangles with two more squares. So we can say that for 6 squares we have $6times4 - 2 times 6 = 12$ triangles. However, I still know that this calculation of mine is quite wrong and based on an awkward thinking. So, what is the correct answer and how?



Thanks for the attention.






combinatorics geometry 3d surfaces polyhedra







share|cite|improve this question























  • Can you now accept my answer? (click the green tick below the score next to my answer.)
    – Parcly Taxel
    1 min ago












up vote
3
down vote

favorite
1









up vote
3
down vote

favorite
1






1





So this is a question that was asked in the International Kangaroo Math Contest 2017. The question is:




The faces of the following polyhedron are either triangles or squares. Each triangle is surrounded by $3$ squares and each square is surrounded by $4$ triangles. If there are $6$ square faces, how many triangular faces are there?






What I did:



Each square shares each of its four neighboring triangles with two more squares. So we can say that for 6 squares we have $6times4 - 2 times 6 = 12$ triangles. However, I still know that this calculation of mine is quite wrong and based on an awkward thinking. So, what is the correct answer and how?



Thanks for the attention.






combinatorics geometry 3d surfaces polyhedra







share|cite|improve this question















So this is a question that was asked in the International Kangaroo Math Contest 2017. The question is:




The faces of the following polyhedron are either triangles or squares. Each triangle is surrounded by $3$ squares and each square is surrounded by $4$ triangles. If there are $6$ square faces, how many triangular faces are there?






What I did:



Each square shares each of its four neighboring triangles with two more squares. So we can say that for 6 squares we have $6times4 - 2 times 6 = 12$ triangles. However, I still know that this calculation of mine is quite wrong and based on an awkward thinking. So, what is the correct answer and how?



Thanks for the attention.






combinatorics geometry 3d surfaces polyhedra




combinatorics geometry 3d surfaces polyhedra






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share|cite|improve this question













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edited 18 mins ago









Parcly Taxel

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37.4k137095










asked 31 mins ago









Faiq Irfan

467217




467217











  • Can you now accept my answer? (click the green tick below the score next to my answer.)
    – Parcly Taxel
    1 min ago
















  • Can you now accept my answer? (click the green tick below the score next to my answer.)
    – Parcly Taxel
    1 min ago















Can you now accept my answer? (click the green tick below the score next to my answer.)
– Parcly Taxel
1 min ago




Can you now accept my answer? (click the green tick below the score next to my answer.)
– Parcly Taxel
1 min ago










1 Answer
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Each edge of the polyhedron is shared between exactly one triangle and exactly one square, as can be inferred from the question statement. Thus, given six squares, there are 24 edges ($6×4$), and thus eight triangles ($24÷3$).



The polyhedron is called a cuboctahedron.






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  • Thanks! I just didn't try thinking it that way....;)
    – Faiq Irfan
    19 mins ago










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1 Answer
1






active

oldest

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active

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active

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up vote
5
down vote













Each edge of the polyhedron is shared between exactly one triangle and exactly one square, as can be inferred from the question statement. Thus, given six squares, there are 24 edges ($6×4$), and thus eight triangles ($24÷3$).



The polyhedron is called a cuboctahedron.






share|cite|improve this answer




















  • Thanks! I just didn't try thinking it that way....;)
    – Faiq Irfan
    19 mins ago














up vote
5
down vote













Each edge of the polyhedron is shared between exactly one triangle and exactly one square, as can be inferred from the question statement. Thus, given six squares, there are 24 edges ($6×4$), and thus eight triangles ($24÷3$).



The polyhedron is called a cuboctahedron.






share|cite|improve this answer




















  • Thanks! I just didn't try thinking it that way....;)
    – Faiq Irfan
    19 mins ago












up vote
5
down vote










up vote
5
down vote









Each edge of the polyhedron is shared between exactly one triangle and exactly one square, as can be inferred from the question statement. Thus, given six squares, there are 24 edges ($6×4$), and thus eight triangles ($24÷3$).



The polyhedron is called a cuboctahedron.






share|cite|improve this answer












Each edge of the polyhedron is shared between exactly one triangle and exactly one square, as can be inferred from the question statement. Thus, given six squares, there are 24 edges ($6×4$), and thus eight triangles ($24÷3$).



The polyhedron is called a cuboctahedron.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 23 mins ago









Parcly Taxel

37.4k137095




37.4k137095











  • Thanks! I just didn't try thinking it that way....;)
    – Faiq Irfan
    19 mins ago
















  • Thanks! I just didn't try thinking it that way....;)
    – Faiq Irfan
    19 mins ago















Thanks! I just didn't try thinking it that way....;)
– Faiq Irfan
19 mins ago




Thanks! I just didn't try thinking it that way....;)
– Faiq Irfan
19 mins ago

















 

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