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What is the most triangles you can make from a capital “H” and 3 straight lines?

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21

4

$begingroup$

So start with an upper case H, and then draw $3$ straight lines. What is the greatest number of closed triangles that you can form? For example:

Note that triangles inside of triangles only count once (e.g. 5 & 6 in the image don't count):

And you aren't allowed to extend the cross line of the H (e.g. 5 in the image doesn't count)

mathematics geometry triangle

share|improve this question

edited 4 hours ago

Hugh

2,28511126

asked yesterday

CurtisCurtis

1086

$endgroup$

add a comment | 

21

4

$begingroup$

So start with an upper case H, and then draw $3$ straight lines. What is the greatest number of closed triangles that you can form? For example:

Note that triangles inside of triangles only count once (e.g. 5 & 6 in the image don't count):

And you aren't allowed to extend the cross line of the H (e.g. 5 in the image doesn't count)

mathematics geometry triangle

share|improve this question

edited 4 hours ago

Hugh

2,28511126

asked yesterday

CurtisCurtis

1086

$endgroup$

add a comment | 

$begingroup$

So start with an upper case H, and then draw $3$ straight lines. What is the greatest number of closed triangles that you can form? For example:

Note that triangles inside of triangles only count once (e.g. 5 & 6 in the image don't count):

And you aren't allowed to extend the cross line of the H (e.g. 5 in the image doesn't count)

mathematics geometry triangle

share|improve this question

edited 4 hours ago

Hugh

2,28511126

asked yesterday

CurtisCurtis

1086

$endgroup$

share|improve this question

edited 4 hours ago

Hugh

2,28511126

asked yesterday

CurtisCurtis

1086

share|improve this question

edited 4 hours ago

Hugh

2,28511126

asked yesterday

CurtisCurtis

1086

edited 4 hours ago

Hugh

2,28511126

edited 4 hours ago

Hugh

2,28511126

asked yesterday

CurtisCurtis

1086

asked yesterday

CurtisCurtis

1086

4 Answers
4

active

oldest

votes

20

$begingroup$

Here's a solution for 7 triangles:

share|improve this answer

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

BassBass

30.3k472186

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You can make $8$ from that picture. See if you can figure it out ;)
  $endgroup$
  – user477343
  12 hours ago
  

  
  
  
  


• 
  
  
  

  
  8
  

  
  
  
  
  
  

  
  $begingroup$
  No, you cannot.
  $endgroup$
  – Bass
  11 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Decrease the angle between three red lines and rotate them ccw slightly. Now two rightmost red lines in the original form will make another triangle with the right pillar of H. According to OP only extending the cross line is not allowed.
  $endgroup$
  – btw
  9 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @btw please draw a picture. The two rightmost red lines (the ones that go through the right edge of the picture) already make a triangle with the right pillar of the H, and hopefully you aren't suggesting that three straight lines could somehow make more than one triangle.
  $endgroup$
  – Bass
  8 hours ago
  

  
  
  
  

add a comment | 

4

$begingroup$

Here's one with six triangles (7 if you count triangles outside of triangles, which you don't):

share|improve this answer

answered yesterday

Brandon_JBrandon_J

1,842230

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I suspect that this is the highest that can be gotten, but I'm not sure.
  $endgroup$
  – Brandon_J
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  My friend alleges that he can get 7, but won't show me how. Considering writing a python script to help prove that 6 is the max. Also I feel like there may be a mathematical proof of this with analytical geometry
  $endgroup$
  – Curtis
  21 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  If you just moved all of your blue lines up or down a bit you’d get 7 won’t you.
  $endgroup$
  – tyobrien
  10 hours ago
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  @tyobrien you'd get one more triangle in the middle, but the two triangles right next to the new one would become quadrangles, right?
  $endgroup$
  – Bass
  10 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Oh yes that would be correct
  $endgroup$
  – tyobrien
  10 hours ago
  

  
  
  
  

 | 
show 1 more comment

1

$begingroup$

Does this count as 8 triangles?

share|improve this answer

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

K SharingK Sharing

13210

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The regions must be triangles.
  $endgroup$
  – noedne
  17 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The question says "Note that triangles inside of triangles only count once", which (I think) is a more complicated way of saying "the triangles are not allowed to overlap".
  $endgroup$
  – Bass
  17 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Oh ok! Thank you
  $endgroup$
  – K Sharing
  13 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Bass Precisely!
  $endgroup$
  – Curtis
  12 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Bass it is possible to create two triangles that overlap, but neither is inside the other, so they're not just different ways of saying the same thing.
  $endgroup$
  – Monty Harder
  11 hours ago
  

  
  
  
  

add a comment | 

-1

$begingroup$

Please have a look at this pic using 3 red lines on letter H.
I count 10 triangles

share|improve this answer

answered 12 hours ago

bi_noobbi_noob

1

New contributor

bi_noob is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  9
  

  
  
  
  
  
  

  
  $begingroup$
  Overlapping triangles are not counted, so there are only 5 in your arrangement.
  $endgroup$
  – Jaap Scherphuis
  12 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yeah, I agree with @JaapScherphuis
  $endgroup$
  – Brandon_J
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @JaapScherphuis Event with overlapping I count 6. How are there 10 here?
  $endgroup$
  – PerpetualJ
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @PerpetualJ I count 10 with overlapping rot13(gbc bs gb u gb zvqqyr yvar gvzrf gjb naq gbc bs u gb obggbz erq yvar gvzrf gjb. ) not sure if that helps
  $endgroup$
  – Yout Ried
  2 hours ago
  
  
  

  
  
  
  

add a comment | 

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20

$begingroup$

Here's a solution for 7 triangles:

share|improve this answer

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

BassBass

30.3k472186

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You can make $8$ from that picture. See if you can figure it out ;)
  $endgroup$
  – user477343
  12 hours ago
  

  
  
  
  


• 
  
  
  

  
  8
  

  
  
  
  
  
  

  
  $begingroup$
  No, you cannot.
  $endgroup$
  – Bass
  11 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Decrease the angle between three red lines and rotate them ccw slightly. Now two rightmost red lines in the original form will make another triangle with the right pillar of H. According to OP only extending the cross line is not allowed.
  $endgroup$
  – btw
  9 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @btw please draw a picture. The two rightmost red lines (the ones that go through the right edge of the picture) already make a triangle with the right pillar of the H, and hopefully you aren't suggesting that three straight lines could somehow make more than one triangle.
  $endgroup$
  – Bass
  8 hours ago
  

  
  
  
  

add a comment | 

20

$begingroup$

Here's a solution for 7 triangles:

share|improve this answer

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

BassBass

30.3k472186

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You can make $8$ from that picture. See if you can figure it out ;)
  $endgroup$
  – user477343
  12 hours ago
  

  
  
  
  


• 
  
  
  

  
  8
  

  
  
  
  
  
  

  
  $begingroup$
  No, you cannot.
  $endgroup$
  – Bass
  11 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Decrease the angle between three red lines and rotate them ccw slightly. Now two rightmost red lines in the original form will make another triangle with the right pillar of H. According to OP only extending the cross line is not allowed.
  $endgroup$
  – btw
  9 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @btw please draw a picture. The two rightmost red lines (the ones that go through the right edge of the picture) already make a triangle with the right pillar of the H, and hopefully you aren't suggesting that three straight lines could somehow make more than one triangle.
  $endgroup$
  – Bass
  8 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Here's a solution for 7 triangles:

share|improve this answer

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

BassBass

30.3k472186

$endgroup$

share|improve this answer

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

BassBass

30.3k472186

edited 12 hours ago

Ahmed Ashour

964312

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

BassBass

30.3k472186

answered 17 hours ago

BassBass

30.3k472186

4

$begingroup$

Here's one with six triangles (7 if you count triangles outside of triangles, which you don't):

share|improve this answer

answered yesterday

Brandon_JBrandon_J

1,842230

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I suspect that this is the highest that can be gotten, but I'm not sure.
  $endgroup$
  – Brandon_J
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  My friend alleges that he can get 7, but won't show me how. Considering writing a python script to help prove that 6 is the max. Also I feel like there may be a mathematical proof of this with analytical geometry
  $endgroup$
  – Curtis
  21 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  If you just moved all of your blue lines up or down a bit you’d get 7 won’t you.
  $endgroup$
  – tyobrien
  10 hours ago
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  @tyobrien you'd get one more triangle in the middle, but the two triangles right next to the new one would become quadrangles, right?
  $endgroup$
  – Bass
  10 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Oh yes that would be correct
  $endgroup$
  – tyobrien
  10 hours ago
  

  
  
  
  

 | 
show 1 more comment

4

$begingroup$

Here's one with six triangles (7 if you count triangles outside of triangles, which you don't):

share|improve this answer

answered yesterday

Brandon_JBrandon_J

1,842230

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I suspect that this is the highest that can be gotten, but I'm not sure.
  $endgroup$
  – Brandon_J
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  My friend alleges that he can get 7, but won't show me how. Considering writing a python script to help prove that 6 is the max. Also I feel like there may be a mathematical proof of this with analytical geometry
  $endgroup$
  – Curtis
  21 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  If you just moved all of your blue lines up or down a bit you’d get 7 won’t you.
  $endgroup$
  – tyobrien
  10 hours ago
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  @tyobrien you'd get one more triangle in the middle, but the two triangles right next to the new one would become quadrangles, right?
  $endgroup$
  – Bass
  10 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Oh yes that would be correct
  $endgroup$
  – tyobrien
  10 hours ago
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

Here's one with six triangles (7 if you count triangles outside of triangles, which you don't):

share|improve this answer

answered yesterday

Brandon_JBrandon_J

1,842230

$endgroup$

share|improve this answer

answered yesterday

Brandon_JBrandon_J

1,842230

answered yesterday

Brandon_JBrandon_J

1,842230

answered yesterday

Brandon_JBrandon_J

1,842230

1

$begingroup$

Does this count as 8 triangles?

share|improve this answer

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

K SharingK Sharing

13210

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The regions must be triangles.
  $endgroup$
  – noedne
  17 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The question says "Note that triangles inside of triangles only count once", which (I think) is a more complicated way of saying "the triangles are not allowed to overlap".
  $endgroup$
  – Bass
  17 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Oh ok! Thank you
  $endgroup$
  – K Sharing
  13 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Bass Precisely!
  $endgroup$
  – Curtis
  12 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Bass it is possible to create two triangles that overlap, but neither is inside the other, so they're not just different ways of saying the same thing.
  $endgroup$
  – Monty Harder
  11 hours ago
  

  
  
  
  

add a comment | 

1

$begingroup$

Does this count as 8 triangles?

share|improve this answer

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

K SharingK Sharing

13210

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The regions must be triangles.
  $endgroup$
  – noedne
  17 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The question says "Note that triangles inside of triangles only count once", which (I think) is a more complicated way of saying "the triangles are not allowed to overlap".
  $endgroup$
  – Bass
  17 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Oh ok! Thank you
  $endgroup$
  – K Sharing
  13 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Bass Precisely!
  $endgroup$
  – Curtis
  12 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Bass it is possible to create two triangles that overlap, but neither is inside the other, so they're not just different ways of saying the same thing.
  $endgroup$
  – Monty Harder
  11 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Does this count as 8 triangles?

share|improve this answer

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

K SharingK Sharing

13210

$endgroup$

share|improve this answer

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

K SharingK Sharing

13210

edited 12 hours ago

Ahmed Ashour

964312

edited 12 hours ago

Ahmed Ashour

964312

answered 17 hours ago

K SharingK Sharing

13210

answered 17 hours ago

K SharingK Sharing

13210

-1

$begingroup$

Please have a look at this pic using 3 red lines on letter H.
I count 10 triangles

share|improve this answer

answered 12 hours ago

bi_noobbi_noob

1

New contributor

bi_noob is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  9
  

  
  
  
  
  
  

  
  $begingroup$
  Overlapping triangles are not counted, so there are only 5 in your arrangement.
  $endgroup$
  – Jaap Scherphuis
  12 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yeah, I agree with @JaapScherphuis
  $endgroup$
  – Brandon_J
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @JaapScherphuis Event with overlapping I count 6. How are there 10 here?
  $endgroup$
  – PerpetualJ
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @PerpetualJ I count 10 with overlapping rot13(gbc bs gb u gb zvqqyr yvar gvzrf gjb naq gbc bs u gb obggbz erq yvar gvzrf gjb. ) not sure if that helps
  $endgroup$
  – Yout Ried
  2 hours ago
  
  
  

  
  
  
  

add a comment | 

-1

$begingroup$

Please have a look at this pic using 3 red lines on letter H.
I count 10 triangles

share|improve this answer

answered 12 hours ago

bi_noobbi_noob

1

New contributor

bi_noob is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  9
  

  
  
  
  
  
  

  
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  Overlapping triangles are not counted, so there are only 5 in your arrangement.
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  – Jaap Scherphuis
  12 hours ago
  

  
  
  
  


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  Yeah, I agree with @JaapScherphuis
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  – Brandon_J
  7 hours ago
  

  
  
  
  


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  @JaapScherphuis Event with overlapping I count 6. How are there 10 here?
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  – PerpetualJ
  4 hours ago
  

  
  
  
  


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  @PerpetualJ I count 10 with overlapping rot13(gbc bs gb u gb zvqqyr yvar gvzrf gjb naq gbc bs u gb obggbz erq yvar gvzrf gjb. ) not sure if that helps
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  – Yout Ried
  2 hours ago
  
  
  

  
  
  
  

add a comment | 

$begingroup$

Please have a look at this pic using 3 red lines on letter H.
I count 10 triangles

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answered 12 hours ago

bi_noobbi_noob

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answered 12 hours ago

bi_noobbi_noob

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bi_noob is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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answered 12 hours ago

bi_noobbi_noob

1

New contributor

bi_noob is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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answered 12 hours ago

bi_noobbi_noob

1