How many triangles are there

How many triangles are there?

The question is how many triangles are there in the following picture?

I have thought to solve it by creating a formula based on the angles of the lines starting from the bottom of each side. I don’t get it right though. Any clues/ideas would be appreciated.

6 Answers 6

But if you are looking for a general formula to count the number of triangles in higher order then check here, (spoiler for the original PE problem).

You can model this using the graph theory. Then make an algorithm that will solve the problem.

Every line that ends connecting more than one line is a node. Find all nodes. Then start from one and look for graphs that contain 3 elements.

I think I found an easier solution for this specific problem without complicating it too much.

We have one basic kind of triangle, and that is the one that our big triangle contains 4 of. Let’s call that order 1.

If we count the triangles inside order 1, we find out that they are 16. They are not so many so we can actually count them manually.

So long we have 16 triangles times 4, 64 triangles.

Now by combining 4 order-1 triangles we get new triangles that originate from the combination.

Lets count the new triangles for ONE side only. Be careful because some of them can be mirrored and some cannot. Actually there is only one that cannot be mirrored.

That gives us 13 new triangles for ONE side only from the combination of order 1 into order 2.

Now we have 13 more for every time we turn the big triangle around. So now we have:

How Many Triangles Are There? Here’s How to Solve the Puzzle

Counting will get you nowhere. Try a little combinatorics instead.

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Unlike the viral “problem,” this puzzle did not hinge on misleading notation. In fact, there’s no notation at all, just a diagram and a question: How many triangles are formed by these six intersecting lines? (The caveat, evident in looking the diagram, is that no two lines are parallel, and there are no points where more than two lines intersect.) The challenge, ideally, was to figure out a procedure to calculate the answer, not to count carefully.

I did not reveal the answer, although I did provide a major hint from Dr. Loh: “A triangle is made of three lines. So if I have six lines, then the whole question becomes how can I pick three lines out of those six lines?”

On Twitter and in the comments on The Times’s website, it was clear that some readers knew immediately how to solve the problem because they had learned combinatorics, an area of math that figures out the number of ways things can be shuffled. Other readers just tried to count, an almost hopeless task. “I don’t think I’ve ever met anybody who’s counted right,” Dr. Loh said.

To solve the problem, start with this observation: Any three lines in the diagram define one and only one triangle. It follows that the total number of triangles will be equal to the number of three-line combinations that can be chosen from a group of six lines.

How do you calculate that? Choose a line, any line. Since there are six lines, there are six choices. Next, choose a line to be the second side of the triangle. At first, you might think there are again six choices, but one line has already been chosen as a line for the first side, so only five choices are left. Likewise, for the third side of the triangle, there are four lines left to choose from.

Thus, the total number of ways you can choose the sides of the triangle equals 6 ×5 ×4, or 120. Clearly, there are not 120 triangles in the diagram. That’s because all of those combinations are being counted more than once.

For clarity, number the lines from 1 to 6, and look at the triangle defined by lines 1, 2 and 3. It’s the same triangle whether you choose line 1, then line 2, then line 3; or line 1, then line 3, then line 2.

Indeed, there are as many ways to create that triangle as there are ways to choose lines 1, 2 and 3. Jumble the digits every which way: 123, 132, 213, 231, 312, 321. There are six possibilities; likewise, any triangle in the diagram could be created six possible ways.

So now, divide out the redundancy. The total number of triangles created by those six lines is (6 ×5 ×4)/6, or 20. That’s the answer.

Here’s where math becomes powerful. The same procedure works for any number of lines. How many triangles are created by seven nonparallel lines? That’s (7 ×6 ×5)/6, or 35. What about 23 lines? (23 ×22 ×21)/6, or 1,771. How about 2,300 lines? That’s (2300 ×2299 ×2298)/6, which is a big number: 2,025,189,100.

The same calculation applies no matter how many lines there are. Compare that approach to brute-force counting, which is not only laborious and error-prone but provides no way to check the answer. Math produces the solution and the rationale for it.

It also reveals that other problems are, at heart, identical. Put balls of six different colors into a bag. Pull out three. How many different possible color combinations are there? 20, of course.

That’s combinatorics, and it’s useful for solving problems of this type. It comes with its own notation, to simplify the process of calculating, and involves a lot of exclamation points. The expression n! — “ n factorial,” when said aloud — describes the product of multiplying all the integers from 1 to n. So 1! equals 1; 2! equals 2 ×1, or 2; 3! equals 3 ×2 ×1, or 6. And so on.

In the problem by Dr. Loh, the calculation for the number of triangles can be rewritten like this: 6!/(3!3!).

It can be written as C(6,3), which is read as “6 choose 3.” More broadly, it’s mathspeak for the number of ways to choose 3 items out of 6. It is generalized into this form:

That’s the equation that students memorize, the useful shortcut. Give it a closer look. The first part — n!/( n– r)! — is what captures the 6 ×5 ×4 in the triangle calculation. The r! is what eliminates the redundancies.

Which is to say, if you forgot the formula, you can reconstruct it by remembering what it’s supposed to solve. Therein lies the beauty of math, or part of it. A good equation is more than a recitation of rules; it expresses why those rules exist, reveals patterns and points to new problems to tackle.

How Many Types of Triangles Are There?

Have You Ever Wondered.

Today’s Wonder of the Day was inspired by taylor. taylor Wonders, “Help on triangles? I’m really confused and struggling to get my math grade up.” Thanks for WONDERing with us, taylor!

Do you know your shapes? Sure, if you see a circle, you probably can identify it, right? But what about those shapes with multiple sides and lots of angles? Those can be a bit more difficult.

How do you go about identifying shapes? Many kids begin by counting the number of sides. For example, if you see a shape with four sides, what would it be? Depending upon the lengths of the sides, you’ve got either a square or a rectangle.

For example, an equilateral triangle has three equal angles (and three equal sides). Each angle in an equilateral triangle has 60º, since three 60º angles would add up to 180º.

If you’ve been counting carefully, you know there’s one more option. If a triangle doesn’t have two or three equal angles, it could have zero equal angles. These triangles, which also don’t have any equal sides, are known as scalene triangles.

Standards : CCSS.MATH.CONTENT.1.GA.1, CCSS.MATH.CONTENT.2.GA.1, CCSS.MATH.CONTENT.4.GA.2, CCRA.L.3, CCRA.L.6, CCRA.R.1, CCRA.R.2, CCRA.R.10, CCRA.SL.1

Wonder Contributors

We’d like to thank:

Miss Cross’ Class from OH
for contributing questions about today’s Wonder topic!

We Spent All Day Arguing About This Triangle Brain Teaser. Can You Solve It?

Mathematicians reveal the real answer. See if you’re right.

There’s nothing quite like a maddening math problem, mind-bending optical illusion, or twisty logic puzzle to halt all productivity in the Popular Mechanics office. We’re curious people by nature, but we also collectively share a stubborn insistence that we’re right, dammit, and so we tend to throw work by the wayside whenever we come upon a problem with several seemingly possible solutions.

This triangle brain teaser isn’t new—shoutout to Popsugar for unearthing it a couple years ago—but based on some shady Internet magic, the tweet below reappeared in my feed today and kick-started a new debate on our staff-wide Slack channel, a place traditionally reserved for workshopping ideas, but instead mostly used for yelling about other stuff that we occasionally turn into content.

Because I’m a masochist, I drew the triangle again and asked everyone on staff to promptly drop what they were doing and attempt to solve the simple question: How many triangles can you find?

I’ll spare you the full conversation—trust me, nobody wants to see that—but the team’s responses ranged all over the place. Some editors saw four triangles. Others saw 12. A few saw 6, 16, 22. Even more saw 18. One wiseguy counted the triangles in the A’s in the question itself, while another seemed to be having an existential crisis: “None of these lines are truly straight, just curves—thus you cannot define any of them as a triangle,” he said. “There are no triangles in this photo. Life has no meaning.”

We then posed the problem to our Instagram followers, whose replies also ran the gamut, from 5 to 14 to 37. While we acknowledge the high probability of trolling here, it’s clear that people respond to the problem many different ways.

I could’ve listened to my colleagues explain their questionable processes all day, but instead, I reached out to several geometry experts to see if we could arrive at a consensus answer. Turns out virtually all of the mathematicians I contacted found the same solution—but not all of them figured it out in the same way.

If you don’t want to know the answer just yet, stop reading and try to solve the problem first. I’ll meet you back here when you’re done.

Hey, that was quick. Ready for the answer? Unlike some viral math problems that are purposely vague and open for interpretation, this one actually does have a slam-dunk, no-doubt-about-it solution, and it’s 18. Let’s hear from some of the geometry experts as to why.

“I would approach this just like one approaches any mathematical problem: reduce it and find structure,” says Sylvester Eriksson-Bique, Ph.D., a postdoctoral fellow with the University of California Los Angeles’s math department.

The only way to form triangles in the figure I drew, Erikkson-Bisque says, is if the top vertex (corner) is part of the triangle. The base of the triangle will then have to be one of the three levels below. “There are three levels, and on each you can choose a base among six different ways. This gives 18, or 3 times 6 triangles.”

Let’s look at the master triangle again.

“It’s convenient to generalize to the case where there are n lines passing through the top vertex, and p horizontal lines,” says Francis Bonahon, Ph.D., a professor of mathematics at the University of Southern California.

In our case, n = 4, and p = 3. Any triangle we find in the drawing should have one top vertex and two others on the same horizontal line, so for each horizontal line, the number of triangles with two vertices on that line is equal to the number of ways we can choose these vertices, Bonahon says—namely the number of ways we can choose two distinct points out of n, or “n choose 2.”

Remember high school math? That’s n(n-1)/2. And since there are p horizontal lines, says Bonahan, this gives p n(n-1)/2 possible triangles. In our case, that’s 3×4(4-1)/ 2=18.

Here’s a handy breakdown of how to find each possible triangle:

Johanna Mangahas, Ph.D., an assistant math professor at the University at Buffalo, also came to 18—first through simple brute-force counting, then through the same crafty combinatorics as above—but admits our triangle brain teaser isn’t quite as cool as this one from Po-Shen Loh, Ph.D., a math professor at Carnegie Mellon University in Pittsburgh, as featured in the New York Times last year:

This one has a slicker mathematical answer, she says, because here, counting triangles is the same thing as counting combinations of three lines chosen out of six [6-choose-3 = (6*5*4)/(3*2*1)].

“In that case, every pair of lines intersects and there are no triple-or-more intersections, so any choice of three always gives a triangle,” says Mangahas. In the picture I sent her, some lines are parallel, so they can’t be part of the same triangle. “If you took the same seven lines and shook them up a bit, probabilistically they’d most likely land like [Loh’s] problem and you’d have more triangles and a similar cute answer.” (For the record: 35.)

Whew. I haven’t shared this new triangle problem with my coworkers yet. But it’s only a matter of time before they discover it—and argue some more.

🚨IMPORTANT UPDATE 1/30/20🚨: Since publishing this story, many, many readers have reached out to let me know that while 18 is indeed an acceptable answer to this problem, it isn’t the only one, due to some unintentional oversight on my part. I could have made this much easier on readers—and, crucially, much easier on my inbox—had I just sketched the triangle on plain, white computer paper. But no.

I unfortunately drew this triangle on lined paper, and lots of smart people have correctly pointed out that, well, actually, if you count the light blue parallel lines in the image in addition to the dark blue lines written in marker, there are actually more than 18 total triangles here—considerably more. I never specified to only use those dark blue lines, and thus, I am wrong. You are right.

One reader, Ralph Linsangan, totally owned me by sending this image, in which he marks every additional triangle found under the technicality, flagging 17 additional triangles for a total of 35. Behold:

That kind of dedication is just one of many reasons I love Popular Mechanics readers. We can’t get anything past you guys. Until the next teaser!

🚨YET ANOTHER TRIANGLE UPDATE 1/31/20🚨: Since posting the last update, I’ve heard from even more of you, continuing to chide me—and your fellow readers—for not considering additional possible triangles. Let’s hear from reader Derek Schneider, who sent in another graphic suggesting there are 45 triangles.

If we follow the original rules however, I count and additional 9 that are definite (in green) and one that could be open to interpretation depending on how you visually place the top vertex (in purple)…I would personally count it.

Reader Poingly, meanwhile, wrote in to say we’ve been making a “grave error” in counting the triangles all along:

Take the bottom right corner, for instance, it shows one arrow for one triangle. However, these light blue lines could conceivably form as many as THREE triangles in this one corner alone:

While some of these MAY be somewhat debatable (ie, where EXACTLY do the light blue lines intersect the dark ones and do they technically form a triangle or a quadrilateral), I have counted SEVEN ADDITIONAL triangles that may be made in this way. This bring the total number of triangles up to 42.

The bad news is that we missed some triangles. The good news is that this confirms that life clearly DOES have meaning, as evidenced by the exact number: 42.

Outstanding point, Poingly. Reader James Goodrich took it another step further, suggesting we open our minds to consider what a triangle could be:

Well, according to your reader, who pointed out 17 additional triangles (using the «Andrew didn’t specify what lines can comprise the 3 edges of a triangle» clause), failed to clearly find quite a lot more. Take, for example, the bottom-left mini-triangle in the 30 Jan 2020 «Important Update» addendum. Would not the areas of the mini-triangle and the area of the rhombus adjacent to it, combined, make for another triangle?

Another idea for consideration: Triangles have 3 angles (who would have guessed?); however, I would postulate that how you describe a triangle, by way of said angles, would generate different triangles. Given a triangle T, with vertices A, B, and C, t-one might indeed be described by ABC, with B being the central angle. I suggest that t-two, being described by BAC, is different. Similarly for BCA.

If we then take a particular case, right-angle triangles, we can derive sine, cosine, and tangent functions (SOH, CAH, TOA). If we were to apply that to the triangle (and relax the right-angle requirement, it might mean that BAC is different than CAB. Of course, exceptions are made for isoscolese and equilateral triangles (the latter would only have 3 distinct triangle definitions).

I haven’t quite thought of how to quantify each suggestion (and applying the latter after the former would increase the count still), so I don’t have an easy number for you to use in an updated important update (if you found my ideas worthwhile to update).

I did, James. And I’ll be waiting. Begrudgingly, I decided to take one last stab at figuring out just how many additional triangles there could be given our new chaotic rules, and arrived at 43, for a total of 61:

I’m quite sure, however, that someone reading this will very quickly tell me I’m wrong yet again and deliver proof of even more hidden triangles, sending me down another rabbit hole on the long and winding path to eventual insanity. (Side note: I haven’t seen my wife in three days. Please tell her I love her.) So I’m issuing one last challenge: If you can find the most possible triangles in the original image, show me your work, and definitively prove your supremacy, I will update this story one final time and crown you the Triangle King or Queen, now and forever. Godspeed.

How many triangles

I saw this question today, it asks how many triangles are in this picture.

I don’t know how to solve this (without counting directly), though I guess it has something to do with some recurrence.

7 Answers 7

Algebra beyond this point corrected.

The total number of downward-pointing triangles is therefore

The final figure, then is

Tabulating numbers

$$ \begin n \backslash k & 1 & 2 & 3 & 4 & 5 & 6 & \Sigma \\\hline 1 & 1, 0 &&&&&& 1 \\ 2 & 3, 1 & 1,0 &&&&& 5 \\ 3 & 6, 3 & 3,0 & 1,0 &&&& 13 \\ 4 & 10, 6 & 6,1 & 3,0 & 1,0 &&& 27 \\ 5 & 15,10 & 10,3 & 6,0 & 3,0 & 1,0 && 48 \\ 6 & 21,15 & 15,6 & 10,1 & 6,0 & 3,0 & 1,0 & 78 \end $$

Finding a pattern

Now look for patterns:

Using these rules, you can extend the table above arbitrarily.

Looking up the sequence

Now take that sum column to OEIS, and you’ll find this to be sequence A002717 which comes with a nice formula:

There is also a comment stating that this sequence describes the

Which sounds just like what you’re asking.

References

If you want to know how to obtain that formula without looking it up, or how to check that formula without simply trusting an encyclopedia, then some of the references given at OEIS will likely help you out: