How many triangles are hidden in the pattern?
What strategy might you use to count them all to ensure you don't miss any out?

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Topics: Starter | Problem Solving | Shape

• Liz, Somerset
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• Provoked lots of discussion, using it with a small group the children came out to draw over shapes found on whiteboard.
• Student:Ben Clark, Year 7 St Peters Dorset
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• It is easy; it is 28. If you count the green and yellow triangles and the halves of each square.
• Year 7, Westbourne House
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• Hugo says its 44 but we all think their might be more!
• 8B, Freebrough Specialist Engineering College
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• We also think there are 44!
• Andy, Grantham
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• There are 44, unless I am very much mistaken.
• Mrs Myers and her Y9 brainboxes!, Woldgate College, Yorkshire
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• We found 44 too!
• Mrs Warner's year 7, Burford School
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• We found 32 .....
• Nam Anh, 7T British International School
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• There are 44!
• P Hillman, Year 10, BIS, Jakarta, Indonesia
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• Sahara found 44. George found 48! we think he has counted the same ones twice.
• 711 , Bay House School, Gosport, Hants
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• 711 found 44 too!
• Evan Gingrich, Mountainview class of '11
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• There are 44. 8 yellow triangles. 8 green triangles. 16 halves of the four little squares. 4 halves of the whole thing. 8 quarters.
• Mrs Johnston's Yr 7 Class, Wardle High School, Rochdale, Lancs
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• The majority of the class found 44. however some people imagined extra and found 48!!
• Cody Robinson, Monticello High School
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• Its easy.. there are 44. Think of it this way.. there are 8 triangles per square. and there are six squares in the figure. That gives you 48 but you have to remember that the tilted center square has 4 triangles that are being used as part of the 8 in the smaller 4 squares. So 4 are counted twice.. 48 - 4 = 44 total triangles.
•
• Found 44 here too.

16 + (8X2) + (4X2) + (1X2) + (1X2)

16= smalls ones
(8X2) = X2 triangles
(4X2) = X4 triangles
(2X2) = half of the square.
• The Red Maids' School, Y7
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• We agree that there are 44.... But it's pretty difficult to prove!
• P7's in Mrs Sweeney's Maths Class, Netherlee Primary School
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• We all really enjoyed this problem and found it fun. We worked it out logically and got the right answer!
• Mr. Hutchinson, Harrytown High School Stockport
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• Abi in 10B2 from Harrytown in Stockport got 40
• Cheltenham Bournside School, year 10
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• We found 44 triangles after much debating.:)
• Mr Cansdale, Hereford Technology School, Grimsby
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• Year 11 think 44
• 1M, St. Matthew's Academy. Scotland
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• It was mind boggling. We thought it was 44 triangles.
• Mr Sinclair, Oban High School, Scotland
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• Mr Sinclair's class at Oban High School, we found 44 after a class discussion.
• Anthony, Maine
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• Anthony and Michael (Daddy) say 44! Hooray! We got 44. What now!
• Oliver Driscoll, Kings Bruton
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• I disagree with Ben Clark. I think there are 44.
• Benjamin Bird, Belchertown MA
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• There are 44. You have 16 small ones. you have 4 for each of the four squares which is 16 more. That's 32. Then you have 4 more big triangles. 36. 4 more bigger triangles. 40. And 4 of the biggest triangles with that makes 44.
• Maria B., Houston, TX
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• I found 48.
They are all right triangles; therefore, if you put 4 together, you can make a square. And within that square there is the possibility of seeing 8 triangles when you make draw an X going from corner to corner.
Count the number of squares in the puzzle (don't forget the overall square, and you'll see 6.
6 times 8 triangles per square = 48.
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• Greg,
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• Actually you are all wrong, there are 48 triangles. No one has accounted for the 4 that head outwards from the centre line and meet in the middle of the outer lines.
• George Bush, Atlanta
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• Greg and Maria, you are wrong. It is only 44. The extra triangles you are suggesting are already accounted for in the 3rd biggest triangle possible. There is only 8 of them.
• CML, New York
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• 44, those who counted 48 are re-counting some of them.
• Antonio, Pittsburgh
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• so, Maria in your method: "counting by squares", when you count the center square, if you add 8 triangles to your count you are actually counting 4 extra triangles, already counted as halves of the smallest squares. I have a better idea, COUNT BY TRIANGLES!!
there are 4 triangle sizes
T size: 16

TT size: 16

TTTT size: 8 (yes, i counted the ones pointing outwards and inwards)

TTTT
TTTT size: 4 (16+16+8+4=44)
• ARP, Chicago, IL
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• The answer is forty-four. The problem with the 'square quarters and halves' method is that quarters of larger squares are in fact the halves of the smaller ones, which leads to 're-counting.' Then, staying true to that method one also has to take into consideration eighths and sixteenths, which leads to even more 're-counting' and one ends up in a big mess. In this case, counting the triangles by size is probably the best method. When one is finished counting the triangles that consist only of themselves, that number (1) is doubled, as they are all right triangles. Now one counts those that consist of two triangles, then four and finally, eight.
• Desiree Finkbeiner, Arkansas
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• I don't care how many people only see 44. You people are still missing 4! There are 48! there are 6 possible squares, each with 8 possible triangles. Do the math, if you divide 44 by 8 you only get 5.5

Some argue that at 48 some are counted twice. I assure you, they are not counted twice. Look again.
• Chris, Melbourne, Australia
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• There are only 44, supposed extra four that are being counted in the second largest square (the one tilted 45 degrees) are already counted within the smallest possible square (the one made up of four little triangles). People trying to use math to solve the problem via the number of squares formed are not accounting for overlap of smaller triangles counted within other squares
• Hans, Los Angeles
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• There are 48, and no I did not count the middle squares triangles twice. If you look opposite of the ones you think we are counting twice, you'll find 4 more triangles that we didn't count because they overlap onto two squares. From corner to corner. Plus adding on the the 4 hypotenuse triangles, that makes 8 in the large one + 8 + 8 + 8 +8 + 8.
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• I personally found 48. There are 16 total small triangles (green and yellow). Next, there are 4 medium-sized triangles in each quarter of the picture (by combining a yellow and green triangle, then rotating around each quarter) for a total of 16 more. Next, there are 4 medium-sized triangles by looking at the middle of the picture and rotating around (again, each one of these is one yellow and one green triangle) for a total of 4 more. 4 large triangles are found if you consider the edges of the picture a base of a large-sized triangle. Also, 4 equally-sized large triangles are found if you take the middle axis of the picture to be triangle bases, with the tips ending at the middle of each picture edge. Lastly, you find 4 super-large triangles by halving the entire picture 4 different ways. This adds up to 48 triangles. Congrats.
• Horatio Hornblower, University at Buffalo
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• Ryan, you are counting four of the triangles twice. The four medium sized triangles that you count by looking at the middle of the picture and rotating around you had already counted as medium sized triangles in each quarter of the picture. So the total is only 44, not 48.
• Mike, Naples, FL
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• 44, you can only count 4 triangles from the middle box.
• Eric, Houston
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• 44. For those who found 48 and are counting 20 medium(1 yellow & 1 green) trianges, it is not possible to have more medium trianges than small trianges.
• Tram Harvey, UK
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• I have found 73 triangles....oh dear, am I wrong ?
• Tram Harvey, UK
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• Excuse me I have just confused with another test when you only count the triangles with the 3 points joined. That's why I gave the answer 73 ! Sorry about that ... Furthermore I have only seen your answer later when I browsed down the page.
• Allison, Pensacola
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• Ok, anyone who listens to the actual answer is wrong. There are 48 triangles not 44. Its really hard to explain but I went over it too many times and there ARE 48 triangles.
• Laura , Los Angeles, CA
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• I agree with 48.
• Eric Mattair, St. Petersburg, FL
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• (anonymous) from Sakuckoinvius, Canada has the right total, but there's actually 4 triangles that take up half the square each
• Ed Coyne, San Angelo TX
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• I got foty-fo.
• Willie Brown II, Houston, TX
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• 44 is the correct answer without recounting existing triangles.

Each small square contains 8 triangles - 8 x 4 = 32

The inner square has 4(top/bottom then left/right) 2 x 2 = 4

The outer square has 8 triangles - 4 inside & 4 outside - 4 + 2 = 8

Total = 44

The "48 count" error comes from recounting the inner triangles on both large squares.

This was fun, hope everyone else enjoyed the mental exercise as well! :)
• Civokrud, KY
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• Eh Dudes, if you scroll down a little you'll see the answer is 44. It's awesome how they do that so we don't have to argue...
• Mike, P-town
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• Since Hans used hypotenuse in his theory to make 48...
these are all right triangles
the hypotenuse is opposite of it's 90 degree angle
one hypotenuse can be shared by two triangles (count "mirror" sides)
a hypotenuse that has
1 line = 16
2 line short = 16
2 line long = 8
3 line = 0
4 line = 4
total of 44

A change in perspective may be beneficial as a teaching aid. we all learned how to count and what shape a triangle was by age 6. Now lets teach common sense.
• Eric, Miami
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• I counted them. Its 44, and the above comments are right concerning re-counting the same triangles in order to get 48. The correct answer is 44.
• Jim, Jimdiegnan@gmail.com
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• Great exercise and it held my family of 35 really concentrating on gettig it right.
• Colleen, NC
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• I have to agree with 48.
Based on the number of small triangles in each triangle, there are:
16 with 1 small triangle
20 with 2 small triangles
8 with 4 small triangles
4 with 8 small triangles.
• Colleen, NC
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• Oops...
I did just notice the overlap of the ones with 2 small triangles. Sorry!
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• Michael, United States Air Force Academy
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• There are only 44. If you don't believe me, scroll down and see they're explanation. Other people have explained it to, but just SCROLL DOWN. its 44.
• Hannah, Ventura, Ca
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• I agree with 44 it is mathematically impossible to have 48. The error is in the smaller triangles running from the center facing outwards. The first time I counted I got 42 but failed to count 2 of the "half-square" triangles.
• Conor Craig, Rehboth Massachusetts
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• People with 48 are double counting 4 of the squares when they look in the middle box that sits on it point... they are counting 4 inside that which are already counted when they look at the 4 quarter sections of the large box. In other words you can count 8 triangles for every box besides the one in the middle... on that one you can only count 4 because it shares 4 with he quarters of the large triangle.
• Kelsey, Mississippi
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• There are 46 triangles. The answer diagram doesn't count the two large tranles that nake up the square. If you don't count the large two then the answer would be 44. Please re-look if you 44 to see the large two triangles.
Need Help? Cut the square in half from the top left corner down to the bottom right. Now do you see the last two triangles?
• Matt S., New Hampshire
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• I'm a thirteen year old. And you know what? I count 44.
I believe there are 20 triangles that DON'T have edges that touch the center.
There are then 24 triangles that DO have edges that touch the center.
• Ruan Legaspi, Las Vegas, NV
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• There are 44 triangles. I wear glasses and I counted them without. Please recount.
• Miguel Maldonado, Groton High School, Groton, New York
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• There are 44, Helps when a math genius like me is also into art. Visualizing takes a lot of effort for some people I understand. I counted all the little 16 squares first, which I hope everyone has gotten that far. (16 unit sized)
Then I moved on to see that there are four squares made up of four little triangles, but no more recounting unit sized triangles, move on to look at the triangles made up of putting two little ones together. There are four more on each of these squares. Look and see that these triangles I discovered have the vertex of the edges of the smallest squares. Here is where people claim that recounting is done. Well I honestly cant find more than 16 triangles made up of two unit triangles. The square in the middle seems to have four more triangles made up of two unit triangles with vertexes on the edges and in the center, but they have already been counted when looking at the ones made up in the four smallest squares. (16 made up of two)
Looking at the square in the middle, there are triangles made up of four. only four of them because they are made up from the vertexes touching the edge of the biggest square. This center square only has these four triangles made up of four unit triangles, BUT those aren't the only ones in this whole problem. There are four more larger triangles made up of four unit sized triangles. Take a look at the sides of the biggest square. The vertexes of these next four triangles are the edges of the biggest square and the very center of the whole thing. Count... these are made up of four unit sized triangles, and we clearly did not disregard these. So there are 8 total, not 16. (8 triangles made up of four)
Now the hunt for triangles made up of 8... the last four. Yes four not 8, if you are recounting here that's really sad. But honestly I don't know where you recounted but lets look at the last 4. Well this is actually quite simple. Look at the biggest square. These last four triangles made up of 8 unit sized triangles have vertexes only on the corners of the big square, obviously only four of them, and if you see more, then feel free to show me please. (4 made up of 8)
16+16+8+4=!!!! what does it equal!?!?! hmm, 48!
ha ha jk, its 44.
• Melanie Wong Age 11, Bournemouth Sch For Girls
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• I got 44 triangles in 1 minute!
• Ricardo Estopier-Ayala, Strake Jesuit College Prep
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• The Correct answer is 44 triangles, and I couldn't be more sure.
Here are my reasons:
1. Let me explain why there are not 48 triangles: There are 6 squares composed of four or more triangles, which means that these 6 squares can be divide into 8 triangles. This would be 48 if some of the triangle weren't being counted twice. The triangles composed of 2 single triangles that are located in the middle(slanted) square are being counted twice. Once in the slanted square and once when counting the triangles in the four small squares. This would alter the equation of solving the number of triangles to be: 6(# of squares)*8(# of triangles produced by each square)-4(# of squares counted twice)= 44
2. You can also break it down like this: There are
4 triangles composed of 8 of the single triangles
8 triangles composed of 4 of the single triangles
16 triangles composed of 2 of the single triangles and
16 single triangles
All of these triangles add up to make 44 triangles.
• Mike Saldana (student) Prodigy, Ilchs
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• Im not here to spam on anybody else's comments, but for each square there are 4 triangles. there are 6 squares, 6x6=48-4+4=48Triangles.
There are 4 triangles that are recounted. thas where the -4 comes from. but you people are forgetting the triangles that are heading outwards. that are not counted in the square. .
• Aaron, Cedar Lake, IN
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• I agree the answer is 44. It's not that math couldn't solve this problem, but your math is flawed if you have 48. Sure there are 6 possible squares. Actually there are 10 possible squares, if you want to look at it that way. So does that mean there are 80 triangles? If you increase the blocks from four to nine you get 120 triangles as your answer.
• 1 St Year, Scotland
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• 44!! Mr Berry's class St luke's high Barrhead.
• Zach Eveler 17, HHS North Carolina
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• I counted 48. If you say I counted some twice, I think you didn't look well enough. Don't just look at the small ones, and don't just look at them one way. You have to see all the possible combination's: diagonal, up, down, left, and right. Trust me, if you have good perception you will find them.
• Michelle Heinrich, Lanier Middle School
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• Ketchup.
• Cody Fife, Ionia Michigan
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• I dont think I am right because I get to 40 and then I cant find 4 more like everyone is saying.
• Ken, NC
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• There are 48 triangles. They are easy enough to count. After reading the comments, I noticed that another person counted the squared to help determine the number of triangles. This is what I did as well, to make sure I had it right. Using the squares to help determine the number of triangles is an excellent use of math to make a complex problem more simple. You don't have to actually count the triangles to determine the answer this way, but you can if you wish, if only to verify your answer to yourself.
• Andres Velasco Y Coll, Nothingxs@gmail.com
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• There's only 44.
4 'squares' of 4 small triangles each: 16.
4 'squares' of 2 medium triangles (each using 2 small triangles) each in 2 separate configurations (separated from diagonal top left to bottom right, separated from diagonal top right to bottom left): 16.
1 small square in center with 4 large triangles (each using 4 small triangles), each large triangle sharing two of its small triangles with 2 other triangles: 4.
1 large square with 4 large triangles (each using 4 small triangles), each large triangle not sharing any of its own small triangles with any of the other triangles: 4.
1 large square of 2 very large triangles (each using 8 small triangles) each in 2 separate configurations (separated from diagonal top left to bottom right, separated from diagonal top right to bottom left): 4.
That's 44 triangles total.
• Brian, West Milford, NJ
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• There are 44. There are indeed 6 squares and each square does have 8 triangles in it, which is what leads a lot of people to confirm what their eyes are telling them and state there are 48. The problem with the math, is that if one looks at the center square (the one that is rotated 45 degrees from the large square and is made up of the eight small triangles that touch the center of the largest square) 4 of the triangles created from that square (the four triangles that are created by dividing the square into 4 equal sized sections made up of 2 triangles each) are already accounted for when one counts the number of triangles made up by the 4 smallest squares.
Said in a different way, 4 or the triangles created when dividing 4 small sqaures into triangles, are the same as 4 of the triangles created when diving the mid-sized (or center square) into quarters.
• Campuschris, Alabama
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• There are 44 total triangles. I only found 40 on my own, but if you look at the bottom of the page the answer is listed.
• J Berry, Erm, Not A School, Found The Page By Chance
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• I got 48, would like to try and prove this one way or another...
• Dexter Churnside, Cherryille, NC
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• I found 48, you ask your self how? easy theres 8 triangles inside each little square times 4 triangles makes (32), nd the big square has half as many as all the little ones put togeter which is half of 32 is (16), and 16+32=?? 48.
• Ken, Providence, RI
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• If you think in three dimensions and you wrap around a single axis, there are 48. Cut and paste on to a paper towel tube to check it out. Okay engineering students, how many would get if you warp this around a single point like a geodesic sphere?
• James Sessions, Crestview, Florida
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• There is only 44... figured this out less than a minute. BUT after reading everyone's comment that stated they found 48 made me take a second look just to recheck... I have analyzed every single section and only found 44. .
• Jordan Beck, Seattle, WA
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• Everyone who counted got 44, most people who used math got 48. A physics teacher once taught me that math isn't perfect. This just proves her right.
I counted 44.
• Tckc, Goatstroking@yahoo.com
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• I'm in AP physics and AP calculus and using math and just counting I got 44 because of common sense...here is my reasoning:
in the smallest squares there are 8 triangles. 8x4=32
in the center square there are 4...but if u count ones you have already counted before then there will be 8. so 32+4=36
then there are 4 enormous ones using half of the square, there are 4 of these as well. 36+4=40
and finally from the main square side, take each one and make a triangle out of it going to the center of the triangle. thus giving you the final 4 to equal 44.
if u get over 44 your wrong. if u get under your missing something. the center square has 16 triangles total but u can only count 4 out of those 16 or you will be recounting from the smallest square.
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• Kong Hang, Minneapolis, Minnesota
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• My first time counting I got 36, after rechecking 2nd time I got 44.
Good puzzle, made me think a bit.
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• Most the kids spotted 44, some said 48 although dont know where they made up an extra 4 triangles!!!!!
• Ms Somas, London
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• My year 7s loved this! The closest a pupil got was 42 but they really did enjoy counting together.
• Year 4, West Sussex
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• Year 4 found 44 triangles in the square!
• Mr Walker's S3 Class, Stromness Academy
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• We also came up with 44 although Robert and Finley insisted on 16 as they reckoned the question didn't explicitly state that you could combine them - a bit pedantic?
• Grade 9A, AGS, Abu Dhabi, UAE
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• Jelena found 44 within just 1 minute and explained her findings clearly to the class. Great fun!
• Ms S Allen, Dundee, Scotland
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• My class of year 6's loved this! Well done to Matthew, Lyly, Geno and Shiv for answering it correctly first! :) It is small activities like this which will really get your brains warmed up for the day, as I tell my class.
• Miss Wisdom And 7A3,
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• 7A3 think 44 as well - we are the best maths class ever!!!!
• Year 8, Welshpool High School, Powys
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• Tristan, Jack,Leanne, Jamie, Chloe and kate got 44. Ryan got 42. Mary got 99!
• Year 5 Mere Green School, Birmingham, UK
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• We got 44! We're aged 9-10!
• Year 5, Monmouthshire
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• Many of us were very close, and one member of our class got 44!
• Yr 6 Eagles, Stanwell Fields
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• We found 44 too!
• Mr Moghul And 7H1, Light Hall School, Shirley
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• My wonderful 7H1 class thought it was.... 44.... and they are spot on!!
• Mr D Smith And Year 8, Bodmin College
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• We did this as a class starter. We found there were 4 of any triangle by symmetry and counted 44 in total.
• Mr Trollgate, Wentworth High School
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• Definitely 48, just counted them up.
• Mr Murden, Hayle Community School, Cornwall
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• We were amazed that there was such a discussion about the answer when it was at the bottom of the page.
• D. Mcphee, Powys
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• Llandrindod Wells High School Year 8
Ben solved it in less than a minute. Followed by Lois, Emlyn, Lucy, Jake and Owain.
• Mrs R, Streethouse
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• Streethouse Y5 and Y6 enjoyed this most of us found 44! Brilliant!!!
• Nuala, Toot Hill 7T6b
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• Triangle sized
(1= I small triangle) The amount of them
1 16
2 16
4 8
8 4
That is a rough total of my answer
When I say size I mean the amount of tiny yellow or green ones, that fit in the triangle.
• Mitat, Kosova
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• There are 44 triangles ... I'm sure 100%.
• 9X1, Wintringham Grimsby
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• 9X1 the creme of the crop think 42 (majority).
• Dylan, Stanwell School
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• Check the bottom its 44 it tells you the answer.
• Edo, Indonesia
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• Every square has 8 triangle (4 really visible, and + 4 using diagonal line as baseline of the triangle).
there are 4 visible small square + 1 big square with the same pattern as the small square, so there are 5 square.
but there is 1 big square rotated 90 degree diagonally, and in this square, we can only count 4 triangle more since the other 4 is already count on small square.
so ...
(5 square * 8 triangle) + 4 = 40 + 4 = 44 ...
• Bushey Meads School, Year 10 Dons
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• 44.
• Jemery Johnston, Albany Ny
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• There 44 total counting all the triangles that you see.
• Andrew Wells, Gwent Wales UK
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• I got 44 in 5 seconds . My methodology was to look at the answers and go with the majority.
• Mr C And 9x2, Yate International Academy
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• We thought 16 then realised there were more, had a big argument then decided that is was 44 after all.
• Year 6, Stamford Green Primary School, Epsom
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• 44- we really enjoyed this problem.:-).
• Ajay, Visakhapatnam, India
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• The large square can be considered as a combination of (1) 4 small squares, (2) 2 vertical rectangles or 2 horizontal rectangles with each rectangle consisting of two adjacent small squares (3) one large square.
Effectively, we have 5 squares and 4 rectangles.
Each square has 4 triangles basing on each side with center point of the square as the top of the triangle. Also each square has two diagonals with two triangles on either side of the diagonal.
This makes eight squares within each square and for five squares, the number of trinagles total 40.
Each rectangle has 2 triangles basing on each of the broader side as the base with center point of the other broader side as the top of the triangle.
The triangles with outer sides are already counted as part of the large square. Thus the four rectangles contain 4 additional triangles.
This makes the total to 44.
(4x4) = X1 triangles(smalls ones)
[ 4 in each small square]
(4X4) = X2 triangles(2 smalls ones as one)
[ 4 in each small square]
(4X1) = X4 triangles(4 smalls ones as one)
[4 in the large square]
(4x1) = x4 triangles(on vertcally and horizontally half of large square)
[4 in the large square]
(4X1) = X8 triangles(diagonally half of the square)
[4 in the large square]
(4x4) + (4X4) + (4X1) + (4X1) + (4X1) = 44.
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• Anywhere from 0 to 72.
0 it is a green and yellow square.
72 if you count all the triangles.
216 if you assign a base and equate from there.
• N.D., Texas
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• Sorry to say, the solution is WRONG! There are at least 48. Take the large outside perimeter box - run 1 diagonal line from corner to corner. That makes 2 larger triangles they never counted. Remove that line and add 1 diagonal line from the opposing corners. That's 2 more for a total of 4 additional triangles that are not included in the solution.
Thank you!
• N.D., Texas
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Hate it when I make myself into an idiot in public.
• Mike, Suffolk UK
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• Well - I may be an odd ball and need glasses but seems to be 40 to me !!!!
• Industrial Engineer, Texas
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• Today was the first time I notice this place on the web. It's clear that the count is 44. Many of the answers above provide the details logically.
There are 16 small size triangles, call it basic, then
there are 16 double size than the basic, (Each = 2 basic), then
there are 8 quadruple size triangles (Each = 4 basic)
And finally, there are 4 Octuple size triangles (Each = 8 basic). In other words, the latest is equl to half of the square.
So adding all gives a count of 44.
• Craig, Kearney Ne.
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• Talk about mind boggling ? there are 44. glad I didnt have to try that in grade school !
• Roman, All Students In Mr Hunt's Class
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• Roman Lucas says it is 44.
• Anastacio Contreras, Mexico
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• Actually, since there were no rules established I simply turned the page over and got 88 triangles!!!
• Z Kotebe, Ethiopia
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• There are 48 triangles all together. Because from each square one can count 8 triangles. there are 6 squares. so 6 X 8 = 48.
• Yem, Ethiopia
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• There are exactly 44 triangles
From the 5 squares, there 40 triangles and only 4 triangles from the middle square 4 triangles.
• Miss Strange, St Martin's Primary School, Guernsey
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• In Year 3 at St. Martin's we thought there were 44 triangles.
• Jeffalextn, Tnnessee
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• I counted so many times, I'm seeing triangle's on my T.V. and my wall's. But, So many 44's out there, I just decided to just go with that. Let's see some more. Everyone learn's by illusion's like that...
• Roger, Somerset, PA
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• 44 triangles... first count...
Basic little triangles: 16
Each half (small squares): 16
Each half (big squares): 4
Each half (4 rectangles)
(2 vert & 2 horiz); 8
Total: 16 + 16 + 4 + 8 = 44.
• Mr Cansdale 9A3, Oasis Wintringham
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• The majority of the class think 34!
• 8A4 Mr Cansdale, Oasis Wintringham
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• The majority of the class think 42, the rest thought lower than 42.
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• (4x4x8)+(3x3x8)+(2x2x8)+(1x2x8)=40.
• 9b MA/6,
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• We found 44. We found the small ones difficult to find but once we got the hang of it , it was easier to find the big ones.
• Robert Felton, San Diego
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• 48, they say the answer to this is 44 but I clearly count 48.
I don't agree with the answer.

If you look at the rectangular part of the diagram, there are 4 rectangles with 2 triangles ea. (These are distinctly different,
they are rectangular shaped triangles. not recounted) = 8

This is in edition to the 2 squares which contain 4 triangles ea.
( The main one & the diamond in the middle, distinct & separate not recounted) = 8
Then there are 4 squares with 8 triangles in ea. (distinct & separate not recounted) = 32
For a total of 48. triangles.
• Robert Felton, SanDiego
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• The people who got 48 are right. There are 6 squares & 4 rectangles in the diagram. the 6 squares are as follows:
The main one: which has - 8 separate triangles = 4
The (diamond shaped) square
in the middle has - 8 separate triangles = 4
The Maim square is divided
into 4 squares that has - 8 separate triangles = 32
which give you 40 triangles. = 40
then there is 4 rectangular
parts of the diagram which
are 2 of the 8 square boxes
side by side.Ea one has
2 rectangular triangles in
ea.( they separate & distinct,
not recounted) for a total of
8 = 8
for a total of = 48
I've noticed that there are many ways of counting the triangles,
largest to smallest, within the the squares / rectangles.
But you cannot get around the triangles in the rectangles they separate triangles of their own & must be counted.
• M Lambert, Woodland Middle School Academy
•
• 44. Year 8 agree.
• Mrs Foster, School For Girls
•
• My class found 16.
• Bay House, 10B7
•
• We also think it is 44, but we still think there might be more!!!!!!!
• Class 5TC, Seoul, South Korea
•
• We enjoyed the investigation. Four of us found 44 triangles.
Thank you!
• Primary 7, Mosspark Primary Glasgow
•
• It was quite tricky but we got it in the end. The answer is 44.
• Hodge Hill Mix, Form 9n5
•
• My class has just come from a snowball warzone and has worked out 44. I was well impressed despite them being frozen from their lunch time labours.
•
• We think 44 or maybe 48.
• Matthew, 3D, Craigslea State School
•
• I found 44 triangles too!
• Mr Eagle, Eagletown
•
• Eagles Class did marvelously well and got it within 5 seconds.
• Mr P, Alleyns
•
• 7RB had a few different ideas - some of which were equal to 44. We enjoyed thinking about how to be systematic.
Thanks.
•
• Ans:44.
• TeriK, VA
•
• If you count the 3D triangles, then you get 48. I say, why not count them?
• AndyGlover, Devon
•
• Two methods = (1^0)+(2^0)+(3^0)+(4^0)+(1^1)+(2^1)+(3^1)+(4^1)+(1^2)+(2^2)+(3^2)+(4^2)= 1+1+1+1+1+2+3+4+1+4+9+16=44
Or look at the squares in the 4 small ones you have 8 triangles then in the diamond (tilted square) you have 8 but 4 overlap with the previous
then in the big one you have another 8 big triangles so (4*8)+(4+8)= 44
Hope this helps :).
• Mary, Oregon
•
• Use your crayons, children. Replicate the large square as many times as you need to allow you to color each size of triangle a different color. Unless you are colorblind, you will only be able to color 44 triangles.
• Yr 6 REB, British School Of Kuwait
•
• We all tried really hard at this one! Most of us enjoyed it! Thankyou! Arwa managed to get 44! She got our housepoint today! Thankyou for this starter! We had fun :-).
• SEA, 9A
•
• Samworth Year 9. We think the answer is 44.
•
• We think its 44!!!!!
• Miss Watson's Class, North Yorks
•
• We think the answer is 44. Easy!
• Year 4L, Seoul Foreign British School
•
• We think it is 44!
• Edmead's Barmy Army, Bay House UK
•
• We think there are 44 triangles in total.
• Apinaya, Swindon
•
• 44 I am so sure we checked the answer.
• 10MEA2015, Wythenshawe
•
• We think its 44 :).
• Maria, California
•
• 40!!!! I just went around and counted them 5 times to make sure.
• Alexis, USA
•
• Well, I got thirty six. I am absolutely sure it is correct as I checked too.
• Hashtag YCIS, Pluto
•
• 44 Triangles is the right answer for this question!!!!!!!
#YCISisawesome, #welikehashtags, #whydidihashtagthis.
• John, Nottingham
•
• Good activity - we got 36.
•

[Transum: Yes, the answers are further down the page for those who have a Transum account.]
• MA, SM
•
• We think George is on the money. We agree with 48 triangles.
•
• We found 34.
• Y10 Isca, Exeter
•
• We found 44.

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.

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 Teacher, do your students have access to computers?Do they have iPads or Laptops in Lessons? Whether your students each have a TabletPC, a Surface or a Mac, this activity lends itself to eLearning (Engaged Learning).

Transum.org/go/?Start=January23

Here is the URL which will take them to a related student activity.

Transum.org/go/?to=HowMany

For Students:

For All:
### Content Structure:
The text passage presents a visual puzzle asking how many triangles are hidden in a pattern. It includes a question, a hint for a strategy, and a series of comments from various individuals and classes, most of whom agree on the answer being 44 triangles. The comments also include some explanations and justifications for their counts.

### Analysis:
single_question_with_answer

1. **mathematical question**: the passage contains a clear mathematical question: "how many triangles are hidden in the pattern?"
2. **answer provided**: the answer is provided in the comments, with a majority consensus on 44 triangles. the answer is also confirmed at the bottom of the page.

### explanation:
- **initial question**: the passage starts with a question asking how many triangles are hidden in the pattern.
- **strategy hint**: it suggests a strategy for counting the triangles: "what strategy might you use to count them all to ensure you don't miss any out?"
- **comments and answers**: the comments section includes various individuals and classes providing their counts, with most agreeing on 44 triangles. some comments provide detailed explanations of how they arrived at their counts, breaking down the triangles by size and position.
- **confirmation**: the correct answer is
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
How many triangles are hidden in the pattern?
What strategy might you use to count them all to ensure you don't miss any out?

0

Topics: Starter | Problem Solving | Shape

• Liz, Somerset
•
• Provoked lots of discussion, using it with a small group the children came out to draw over shapes found on whiteboard.
• Student:Ben Clark, Year 7 St Peters Dorset
•
• It is easy; it is 28. If you count the green and yellow triangles and the halves of each square.
• Year 7, Westbourne House
•
• Hugo says its 44 but we all think their might be more!
• 8B, Freebrough Specialist Engineering College
•
• We also think there are 44!
• Andy, Grantham
•
• There are 44, unless I am very much mistaken.
• Mrs Myers and her Y9 brainboxes!, Woldgate College, Yorkshire
•
• We found 44 too!
• Mrs Warner's year 7, Burford School
•
• We found 32 .....
• Nam Anh, 7T British International School
•
• There are 44!
• P Hillman, Year 10, BIS, Jakarta, Indonesia
•
• Sahara found 44. George found 48! we think he has counted the same ones twice.
• 711 , Bay House School, Gosport, Hants
•
• 711 found 44 too!
• Evan Gingrich, Mountainview class of '11
•
• There are 44. 8 yellow triangles. 8 green triangles. 16 halves of the four little squares. 4 halves of the whole thing. 8 quarters.
• Mrs Johnston's Yr 7 Class, Wardle High School, Rochdale, Lancs
•
• The majority of the class found 44. however some people imagined extra and found 48!!
• Cody Robinson, Monticello High School
•
• Its easy.. there are 44. Think of it this way.. there are 8 triangles per square. and there are six squares in the figure. That gives you 48 but you have to remember that the tilted center square has 4 triangles that are being used as part of the 8 in the smaller 4 squares. So 4 are counted twice.. 48 - 4 = 44 total triangles.
•
• Found 44 here too.

16 + (8X2) + (4X2) + (1X2) + (1X2)

16= smalls ones
(8X2) = X2 triangles
(4X2) = X4 triangles
(2X2) = half of the square.
• The Red Maids' School, Y7
•
• We agree that there are 44.... But it's pretty difficult to prove!
• P7's in Mrs Sweeney's Maths Class, Netherlee Primary School
•
• We all really enjoyed this problem and found it fun. We worked it out logically and got the right answer!
• Mr. Hutchinson, Harrytown High School Stockport
•
• Abi in 10B2 from Harrytown in Stockport got 40
• Cheltenham Bournside School, year 10
•
• We found 44 triangles after much debating.:)
• Mr Cansdale, Hereford Technology School, Grimsby
•
• Year 11 think 44
• 1M, St. Matthew's Academy. Scotland
•
• It was mind boggling. We thought it was 44 triangles.
• Mr Sinclair, Oban High School, Scotland
•
• Mr Sinclair's class at Oban High School, we found 44 after a class discussion.
• Anthony, Maine
•
• Anthony and Michael (Daddy) say 44! Hooray! We got 44. What now!
• Oliver Driscoll, Kings Bruton
•
• I disagree with Ben Clark. I think there are 44.
• Benjamin Bird, Belchertown MA
•
• There are 44. You have 16 small ones. you have 4 for each of the four squares which is 16 more. That's 32. Then you have 4 more big triangles. 36. 4 more bigger triangles. 40. And 4 of the biggest triangles with that makes 44.
• Maria B., Houston, TX
•
• I found 48.
They are all right triangles; therefore, if you put 4 together, you can make a square. And within that square there is the possibility of seeing 8 triangles when you make draw an X going from corner to corner.
Count the number of squares in the puzzle (don't forget the overall square, and you'll see 6.
6 times 8 triangles per square = 48.
•

• Greg,
•
• Actually you are all wrong, there are 48 triangles. No one has accounted for the 4 that head outwards from the centre line and meet in the middle of the outer lines.
• George Bush, Atlanta
•
• Greg and Maria, you are wrong. It is only 44. The extra triangles you are suggesting are already accounted for in the 3rd biggest triangle possible. There is only 8 of them.
• CML, New York
•
• 44, those who counted 48 are re-counting some of them.
• Antonio, Pittsburgh
•
• so, Maria in your method: "counting by squares", when you count the center square, if you add 8 triangles to your count you are actually counting 4 extra triangles, already counted as halves of the smallest squares. I have a better idea, COUNT BY TRIANGLES!!
there are 4 triangle sizes
T size: 16

TT size: 16

TTTT size: 8 (yes, i counted the ones pointing outwards and inwards)

TTTT
TTTT size: 4 (16+16+8+4=44)
• ARP, Chicago, IL
•
• The answer is forty-four. The problem with the 'square quarters and halves' method is that quarters of larger squares are in fact the halves of the smaller ones, which leads to 're-counting.' Then, staying true to that method one also has to take into consideration eighths and sixteenths, which leads to even more 're-counting' and one ends up in a big mess. In this case, counting the triangles by size is probably the best method. When one is finished counting the triangles that consist only of themselves, that number (1) is doubled, as they are all right triangles. Now one counts those that consist of two triangles, then four and finally, eight.
• Desiree Finkbeiner, Arkansas
•
• I don't care how many people only see 44. You people are still missing 4! There are 48! there are 6 possible squares, each with 8 possible triangles. Do the math, if you divide 44 by 8 you only get 5.5

Some argue that at 48 some are counted twice. I assure you, they are not counted twice. Look again.
• Chris, Melbourne, Australia
•
• There are only 44, supposed extra four that are being counted in the second largest square (the one tilted 45 degrees) are already counted within the smallest possible square (the one made up of four little triangles). People trying to use math to solve the problem via the number of squares formed are not accounting for overlap of smaller triangles counted within other squares
• Hans, Los Angeles
•
• There are 48, and no I did not count the middle squares triangles twice. If you look opposite of the ones you think we are counting twice, you'll find 4 more triangles that we didn't count because they overlap onto two squares. From corner to corner. Plus adding on the the 4 hypotenuse triangles, that makes 8 in the large one + 8 + 8 + 8 +8 + 8.
•
• I personally found 48. There are 16 total small triangles (green and yellow). Next, there are 4 medium-sized triangles in each quarter of the picture (by combining a yellow and green triangle, then rotating around each quarter) for a total of 16 more. Next, there are 4 medium-sized triangles by looking at the middle of the picture and rotating around (again, each one of these is one yellow and one green triangle) for a total of 4 more. 4 large triangles are found if you consider the edges of the picture a base of a large-sized triangle. Also, 4 equally-sized large triangles are found if you take the middle axis of the picture to be triangle bases, with the tips ending at the middle of each picture edge. Lastly, you find 4 super-large triangles by halving the entire picture 4 different ways. This adds up to 48 triangles. Congrats.
• Horatio Hornblower, University at Buffalo
•
• Ryan, you are counting four of the triangles twice. The four medium sized triangles that you count by looking at the middle of the picture and rotating around you had already counted as medium sized triangles in each quarter of the picture. So the total is only 44, not 48.
• Mike, Naples, FL
•
• 44, you can only count 4 triangles from the middle box.
• Eric, Houston
•
• 44. For those who found 48 and are counting 20 medium(1 yellow & 1 green) trianges, it is not possible to have more medium trianges than small trianges.
• Tram Harvey, UK
•
• I have found 73 triangles....oh dear, am I wrong ?
• Tram Harvey, UK
•
• Excuse me I have just confused with another test when you only count the triangles with the 3 points joined. That's why I gave the answer 73 ! Sorry about that ... Furthermore I have only seen your answer later when I browsed down the page.
• Allison, Pensacola
•
• Ok, anyone who listens to the actual answer is wrong. There are 48 triangles not 44. Its really hard to explain but I went over it too many times and there ARE 48 triangles.
• Laura , Los Angeles, CA
•
• I agree with 48.
• Eric Mattair, St. Petersburg, FL
•
• (anonymous) from Sakuckoinvius, Canada has the right total, but there's actually 4 triangles that take up half the square each
• Ed Coyne, San Angelo TX
•
• I got foty-fo.
• Willie Brown II, Houston, TX
•
• 44 is the correct answer without recounting existing triangles.

Each small square contains 8 triangles - 8 x 4 = 32

The inner square has 4(top/bottom then left/right) 2 x 2 = 4

The outer square has 8 triangles - 4 inside & 4 outside - 4 + 2 = 8

Total = 44

The "48 count" error comes from recounting the inner triangles on both large squares.

This was fun, hope everyone else enjoyed the mental exercise as well! :)
• Civokrud, KY
•
• Eh Dudes, if you scroll down a little you'll see the answer is 44. It's awesome how they do that so we don't have to argue...
• Mike, P-town
•
• Since Hans used hypotenuse in his theory to make 48...
these are all right triangles
the hypotenuse is opposite of it's 90 degree angle
one hypotenuse can be shared by two triangles (count "mirror" sides)
a hypotenuse that has
1 line = 16
2 line short = 16
2 line long = 8
3 line = 0
4 line = 4
total of 44

A change in perspective may be beneficial as a teaching aid. we all learned how to count and what shape a triangle was by age 6. Now lets teach common sense.
• Eric, Miami
•
• I counted them. Its 44, and the above comments are right concerning re-counting the same triangles in order to get 48. The correct answer is 44.
• Jim, Jimdiegnan@gmail.com
•
• Great exercise and it held my family of 35 really concentrating on gettig it right.
• Colleen, NC
•
• I have to agree with 48.
Based on the number of small triangles in each triangle, there are:
16 with 1 small triangle
20 with 2 small triangles
8 with 4 small triangles
4 with 8 small triangles.
• Colleen, NC
•
• Oops...
I did just notice the overlap of the ones with 2 small triangles. Sorry!
•

• Michael, United States Air Force Academy
•
• There are only 44. If you don't believe me, scroll down and see they're explanation. Other people have explained it to, but just SCROLL DOWN. its 44.
• Hannah, Ventura, Ca
•
• I agree with 44 it is mathematically impossible to have 48. The error is in the smaller triangles running from the center facing outwards. The first time I counted I got 42 but failed to count 2 of the "half-square" triangles.
• Conor Craig, Rehboth Massachusetts
•
• People with 48 are double counting 4 of the squares when they look in the middle box that sits on it point... they are counting 4 inside that which are already counted when they look at the 4 quarter sections of the large box. In other words you can count 8 triangles for every box besides the one in the middle... on that one you can only count 4 because it shares 4 with he quarters of the large triangle.
• Kelsey, Mississippi
•
• There are 46 triangles. The answer diagram doesn't count the two large tranles that nake up the square. If you don't count the large two then the answer would be 44. Please re-look if you 44 to see the large two triangles.
Need Help? Cut the square in half from the top left corner down to the bottom right. Now do you see the last two triangles?
• Matt S., New Hampshire
•
• I'm a thirteen year old. And you know what? I count 44.
I believe there are 20 triangles that DON'T have edges that touch the center.
There are then 24 triangles that DO have edges that touch the center.
• Ruan Legaspi, Las Vegas, NV
•
• There are 44 triangles. I wear glasses and I counted them without. Please recount.
• Miguel Maldonado, Groton High School, Groton, New York
•
• There are 44, Helps when a math genius like me is also into art. Visualizing takes a lot of effort for some people I understand. I counted all the little 16 squares first, which I hope everyone has gotten that far. (16 unit sized)
Then I moved on to see that there are four squares made up of four little triangles, but no more recounting unit sized triangles, move on to look at the triangles made up of putting two little ones together. There are four more on each of these squares. Look and see that these triangles I discovered have the vertex of the edges of the smallest squares. Here is where people claim that recounting is done. Well I honestly cant find more than 16 triangles made up of two unit triangles. The square in the middle seems to have four more triangles made up of two unit triangles with vertexes on the edges and in the center, but they have already been counted when looking at the ones made up in the four smallest squares. (16 made up of two)
Looking at the square in the middle, there are triangles made up of four. only four of them because they are made up from the vertexes touching the edge of the biggest square. This center square only has these four triangles made up of four unit triangles, BUT those aren't the only ones in this whole problem. There are four more larger triangles made up of four unit sized triangles. Take a look at the sides of the biggest square. The vertexes of these next four triangles are the edges of the biggest square and the very center of the whole thing. Count... these are made up of four unit sized triangles, and we clearly did not disregard these. So there are 8 total, not 16. (8 triangles made up of four)
Now the hunt for triangles made up of 8... the last four. Yes four not 8, if you are recounting here that's really sad. But honestly I don't know where you recounted but lets look at the last 4. Well this is actually quite simple. Look at the biggest square. These last four triangles made up of 8 unit sized triangles have vertexes only on the corners of the big square, obviously only four of them, and if you see more, then feel free to show me please. (4 made up of 8)
16+16+8+4=!!!! what does it equal!?!?! hmm, 48!
ha ha jk, its 44.
• Melanie Wong Age 11, Bournemouth Sch For Girls
•
• I got 44 triangles in 1 minute!
• Ricardo Estopier-Ayala, Strake Jesuit College Prep
•
• The Correct answer is 44 triangles, and I couldn't be more sure.
Here are my reasons:
1. Let me explain why there are not 48 triangles: There are 6 squares composed of four or more triangles, which means that these 6 squares can be divide into 8 triangles. This would be 48 if some of the triangle weren't being counted twice. The triangles composed of 2 single triangles that are located in the middle(slanted) square are being counted twice. Once in the slanted square and once when counting the triangles in the four small squares. This would alter the equation of solving the number of triangles to be: 6(# of squares)*8(# of triangles produced by each square)-4(# of squares counted twice)= 44
2. You can also break it down like this: There are
4 triangles composed of 8 of the single triangles
8 triangles composed of 4 of the single triangles
16 triangles composed of 2 of the single triangles and
16 single triangles
All of these triangles add up to make 44 triangles.
• Mike Saldana (student) Prodigy, Ilchs
•
• Im not here to spam on anybody else's comments, but for each square there are 4 triangles. there are 6 squares, 6x6=48-4+4=48Triangles.
There are 4 triangles that are recounted. thas where the -4 comes from. but you people are forgetting the triangles that are heading outwards. that are not counted in the square. .
• Aaron, Cedar Lake, IN
•
• I agree the answer is 44. It's not that math couldn't solve this problem, but your math is flawed if you have 48. Sure there are 6 possible squares. Actually there are 10 possible squares, if you want to look at it that way. So does that mean there are 80 triangles? If you increase the blocks from four to nine you get 120 triangles as your answer.
• 1 St Year, Scotland
•
• 44!! Mr Berry's class St luke's high Barrhead.
• Zach Eveler 17, HHS North Carolina
•
• I counted 48. If you say I counted some twice, I think you didn't look well enough. Don't just look at the small ones, and don't just look at them one way. You have to see all the possible combination's: diagonal, up, down, left, and right. Trust me, if you have good perception you will find them.
• Michelle Heinrich, Lanier Middle School
•
• Ketchup.
• Cody Fife, Ionia Michigan
•
• I dont think I am right because I get to 40 and then I cant find 4 more like everyone is saying.
• Ken, NC
•
• There are 48 triangles. They are easy enough to count. After reading the comments, I noticed that another person counted the squared to help determine the number of triangles. This is what I did as well, to make sure I had it right. Using the squares to help determine the number of triangles is an excellent use of math to make a complex problem more simple. You don't have to actually count the triangles to determine the answer this way, but you can if you wish, if only to verify your answer to yourself.
• Andres Velasco Y Coll, Nothingxs@gmail.com
•
• There's only 44.
4 'squares' of 4 small triangles each: 16.
4 'squares' of 2 medium triangles (each using 2 small triangles) each in 2 separate configurations (separated from diagonal top left to bottom right, separated from diagonal top right to bottom left): 16.
1 small square in center with 4 large triangles (each using 4 small triangles), each large triangle sharing two of its small triangles with 2 other triangles: 4.
1 large square with 4 large triangles (each using 4 small triangles), each large triangle not sharing any of its own small triangles with any of the other triangles: 4.
1 large square of 2 very large triangles (each using 8 small triangles) each in 2 separate configurations (separated from diagonal top left to bottom right, separated from diagonal top right to bottom left): 4.
That's 44 triangles total.
• Brian, West Milford, NJ
•
• There are 44. There are indeed 6 squares and each square does have 8 triangles in it, which is what leads a lot of people to confirm what their eyes are telling them and state there are 48. The problem with the math, is that if one looks at the center square (the one that is rotated 45 degrees from the large square and is made up of the eight small triangles that touch the center of the largest square) 4 of the triangles created from that square (the four triangles that are created by dividing the square into 4 equal sized sections made up of 2 triangles each) are already accounted for when one counts the number of triangles made up by the 4 smallest squares.
Said in a different way, 4 or the triangles created when dividing 4 small sqaures into triangles, are the same as 4 of the triangles created when diving the mid-sized (or center square) into quarters.
• Campuschris, Alabama
•
• There are 44 total triangles. I only found 40 on my own, but if you look at the bottom of the page the answer is listed.
• J Berry, Erm, Not A School, Found The Page By Chance
•
• I got 48, would like to try and prove this one way or another...
• Dexter Churnside, Cherryille, NC
•
• I found 48, you ask your self how? easy theres 8 triangles inside each little square times 4 triangles makes (32), nd the big square has half as many as all the little ones put togeter which is half of 32 is (16), and 16+32=?? 48.
• Ken, Providence, RI
•
• If you think in three dimensions and you wrap around a single axis, there are 48. Cut and paste on to a paper towel tube to check it out. Okay engineering students, how many would get if you warp this around a single point like a geodesic sphere?
• James Sessions, Crestview, Florida
•
• There is only 44... figured this out less than a minute. BUT after reading everyone's comment that stated they found 48 made me take a second look just to recheck... I have analyzed every single section and only found 44. .
• Jordan Beck, Seattle, WA
•
• Everyone who counted got 44, most people who used math got 48. A physics teacher once taught me that math isn't perfect. This just proves her right.
I counted 44.
• Tckc, Goatstroking@yahoo.com
•
• I'm in AP physics and AP calculus and using math and just counting I got 44 because of common sense...here is my reasoning:
in the smallest squares there are 8 triangles. 8x4=32
in the center square there are 4...but if u count ones you have already counted before then there will be 8. so 32+4=36
then there are 4 enormous ones using half of the square, there are 4 of these as well. 36+4=40
and finally from the main square side, take each one and make a triangle out of it going to the center of the triangle. thus giving you the final 4 to equal 44.
if u get over 44 your wrong. if u get under your missing something. the center square has 16 triangles total but u can only count 4 out of those 16 or you will be recounting from the smallest square.
•

• Kong Hang, Minneapolis, Minnesota
•
• My first time counting I got 36, after rechecking 2nd time I got 44.
Good puzzle, made me think a bit.
•
• Most the kids spotted 44, some said 48 although dont know where they made up an extra 4 triangles!!!!!
• Ms Somas, London
•
• My year 7s loved this! The closest a pupil got was 42 but they really did enjoy counting together.
• Year 4, West Sussex
•
• Year 4 found 44 triangles in the square!
• Mr Walker's S3 Class, Stromness Academy
•
• We also came up with 44 although Robert and Finley insisted on 16 as they reckoned the question didn't explicitly state that you could combine them - a bit pedantic?
• Grade 9A, AGS, Abu Dhabi, UAE
•
• Jelena found 44 within just 1 minute and explained her findings clearly to the class. Great fun!
• Ms S Allen, Dundee, Scotland
•
• My class of year 6's loved this! Well done to Matthew, Lyly, Geno and Shiv for answering it correctly first! :) It is small activities like this which will really get your brains warmed up for the day, as I tell my class.
• Miss Wisdom And 7A3,
•
• 7A3 think 44 as well - we are the best maths class ever!!!!
• Year 8, Welshpool High School, Powys
•
• Tristan, Jack,Leanne, Jamie, Chloe and kate got 44. Ryan got 42. Mary got 99!
• Year 5 Mere Green School, Birmingham, UK
•
• We got 44! We're aged 9-10!
• Year 5, Monmouthshire
•
• Many of us were very close, and one member of our class got 44!
• Yr 6 Eagles, Stanwell Fields
•
• We found 44 too!
• Mr Moghul And 7H1, Light Hall School, Shirley
•
• My wonderful 7H1 class thought it was.... 44.... and they are spot on!!
• Mr D Smith And Year 8, Bodmin College
•
• We did this as a class starter. We found there were 4 of any triangle by symmetry and counted 44 in total.
• Mr Trollgate, Wentworth High School
•
• Definitely 48, just counted them up.
• Mr Murden, Hayle Community School, Cornwall
•
• We were amazed that there was such a discussion about the answer when it was at the bottom of the page.
• D. Mcphee, Powys
•
• Llandrindod Wells High School Year 8
Ben solved it in less than a minute. Followed by Lois, Emlyn, Lucy, Jake and Owain.
• Mrs R, Streethouse
•
• Streethouse Y5 and Y6 enjoyed this most of us found 44! Brilliant!!!
• Nuala, Toot Hill 7T6b
•
• Triangle sized
(1= I small triangle) The amount of them
1 16
2 16
4 8
8 4
That is a rough total of my answer
When I say size I mean the amount of tiny yellow or green ones, that fit in the triangle.
• Mitat, Kosova
•
• There are 44 triangles ... I'm sure 100%.
• 9X1, Wintringham Grimsby
•
• 9X1 the creme of the crop think 42 (majority).
• Dylan, Stanwell School
•
• Check the bottom its 44 it tells you the answer.
• Edo, Indonesia
•
• Every square has 8 triangle (4 really visible, and + 4 using diagonal line as baseline of the triangle).
there are 4 visible small square + 1 big square with the same pattern as the small square, so there are 5 square.
but there is 1 big square rotated 90 degree diagonally, and in this square, we can only count 4 triangle more since the other 4 is already count on small square.
so ...
(5 square * 8 triangle) + 4 = 40 + 4 = 44 ...
• Bushey Meads School, Year 10 Dons
•
• 44.
• Jemery Johnston, Albany Ny
•
• There 44 total counting all the triangles that you see.
• Andrew Wells, Gwent Wales UK
•
• I got 44 in 5 seconds . My methodology was to look at the answers and go with the majority.
• Mr C And 9x2, Yate International Academy
•
• We thought 16 then realised there were more, had a big argument then decided that is was 44 after all.
• Year 6, Stamford Green Primary School, Epsom
•
• 44- we really enjoyed this problem.:-).
• Ajay, Visakhapatnam, India
•
• The large square can be considered as a combination of (1) 4 small squares, (2) 2 vertical rectangles or 2 horizontal rectangles with each rectangle consisting of two adjacent small squares (3) one large square.
Effectively, we have 5 squares and 4 rectangles.
Each square has 4 triangles basing on each side with center point of the square as the top of the triangle. Also each square has two diagonals with two triangles on either side of the diagonal.
This makes eight squares within each square and for five squares, the number of trinagles total 40.
Each rectangle has 2 triangles basing on each of the broader side as the base with center point of the other broader side as the top of the triangle.
The triangles with outer sides are already counted as part of the large square. Thus the four rectangles contain 4 additional triangles.
This makes the total to 44.
(4x4) = X1 triangles(smalls ones)
[ 4 in each small square]
(4X4) = X2 triangles(2 smalls ones as one)
[ 4 in each small square]
(4X1) = X4 triangles(4 smalls ones as one)
[4 in the large square]
(4x1) = x4 triangles(on vertcally and horizontally half of large square)
[4 in the large square]
(4X1) = X8 triangles(diagonally half of the square)
[4 in the large square]
(4x4) + (4X4) + (4X1) + (4X1) + (4X1) = 44.
•
• Anywhere from 0 to 72.
0 it is a green and yellow square.
72 if you count all the triangles.
216 if you assign a base and equate from there.
• N.D., Texas
•
• Sorry to say, the solution is WRONG! There are at least 48. Take the large outside perimeter box - run 1 diagonal line from corner to corner. That makes 2 larger triangles they never counted. Remove that line and add 1 diagonal line from the opposing corners. That's 2 more for a total of 4 additional triangles that are not included in the solution.
Thank you!
• N.D., Texas
•
Hate it when I make myself into an idiot in public.
• Mike, Suffolk UK
•
• Well - I may be an odd ball and need glasses but seems to be 40 to me !!!!
• Industrial Engineer, Texas
•
• Today was the first time I notice this place on the web. It's clear that the count is 44. Many of the answers above provide the details logically.
There are 16 small size triangles, call it basic, then
there are 16 double size than the basic, (Each = 2 basic), then
there are 8 quadruple size triangles (Each = 4 basic)
And finally, there are 4 Octuple size triangles (Each = 8 basic). In other words, the latest is equl to half of the square.
So adding all gives a count of 44.
• Craig, Kearney Ne.
•
• Talk about mind boggling ? there are 44. glad I didnt have to try that in grade school !
• Roman, All Students In Mr Hunt's Class
•
• Roman Lucas says it is 44.
• Anastacio Contreras, Mexico
•
• Actually, since there were no rules established I simply turned the page over and got 88 triangles!!!
• Z Kotebe, Ethiopia
•
• There are 48 triangles all together. Because from each square one can count 8 triangles. there are 6 squares. so 6 X 8 = 48.
• Yem, Ethiopia
•
• There are exactly 44 triangles
From the 5 squares, there 40 triangles and only 4 triangles from the middle square 4 triangles.
• Miss Strange, St Martin's Primary School, Guernsey
•
• In Year 3 at St. Martin's we thought there were 44 triangles.
• Jeffalextn, Tnnessee
•
• I counted so many times, I'm seeing triangle's on my T.V. and my wall's. But, So many 44's out there, I just decided to just go with that. Let's see some more. Everyone learn's by illusion's like that...
• Roger, Somerset, PA
•
• 44 triangles... first count...
Basic little triangles: 16
Each half (small squares): 16
Each half (big squares): 4
Each half (4 rectangles)
(2 vert & 2 horiz); 8
Total: 16 + 16 + 4 + 8 = 44.
• Mr Cansdale 9A3, Oasis Wintringham
•
• The majority of the class think 34!
• 8A4 Mr Cansdale, Oasis Wintringham
•
• The majority of the class think 42, the rest thought lower than 42.
•
• (4x4x8)+(3x3x8)+(2x2x8)+(1x2x8)=40.
• 9b MA/6,
•
• We found 44. We found the small ones difficult to find but once we got the hang of it , it was easier to find the big ones.
• Robert Felton, San Diego
•
• 48, they say the answer to this is 44 but I clearly count 48.
I don't agree with the answer.

If you look at the rectangular part of the diagram, there are 4 rectangles with 2 triangles ea. (These are distinctly different,
they are rectangular shaped triangles. not recounted) = 8

This is in edition to the 2 squares which contain 4 triangles ea.
( The main one & the diamond in the middle, distinct & separate not recounted) = 8
Then there are 4 squares with 8 triangles in ea. (distinct & separate not recounted) = 32
For a total of 48. triangles.
• Robert Felton, SanDiego
•
• The people who got 48 are right. There are 6 squares & 4 rectangles in the diagram. the 6 squares are as follows:
The main one: which has - 8 separate triangles = 4
The (diamond shaped) square
in the middle has - 8 separate triangles = 4
The Maim square is divided
into 4 squares that has - 8 separate triangles = 32
which give you 40 triangles. = 40
then there is 4 rectangular
parts of the diagram which
are 2 of the 8 square boxes
side by side.Ea one has
2 rectangular triangles in
ea.( they separate & distinct,
not recounted) for a total of
8 = 8
for a total of = 48
I've noticed that there are many ways of counting the triangles,
largest to smallest, within the the squares / rectangles.
But you cannot get around the triangles in the rectangles they separate triangles of their own & must be counted.
• M Lambert, Woodland Middle School Academy
•
• 44. Year 8 agree.
• Mrs Foster, School For Girls
•
• My class found 16.
• Bay House, 10B7
•
• We also think it is 44, but we still think there might be more!!!!!!!
• Class 5TC, Seoul, South Korea
•
• We enjoyed the investigation. Four of us found 44 triangles.
Thank you!
• Primary 7, Mosspark Primary Glasgow
•
• It was quite tricky but we got it in the end. The answer is 44.
• Hodge Hill Mix, Form 9n5
•
• My class has just come from a snowball warzone and has worked out 44. I was well impressed despite them being frozen from their lunch time labours.
•
• We think 44 or maybe 48.
• Matthew, 3D, Craigslea State School
•
• I found 44 triangles too!
• Mr Eagle, Eagletown
•
• Eagles Class did marvelously well and got it within 5 seconds.
• Mr P, Alleyns
•
• 7RB had a few different ideas - some of which were equal to 44. We enjoyed thinking about how to be systematic.
Thanks.
•
• Ans:44.
• TeriK, VA
•
• If you count the 3D triangles, then you get 48. I say, why not count them?
• AndyGlover, Devon
•
• Two methods = (1^0)+(2^0)+(3^0)+(4^0)+(1^1)+(2^1)+(3^1)+(4^1)+(1^2)+(2^2)+(3^2)+(4^2)= 1+1+1+1+1+2+3+4+1+4+9+16=44
Or look at the squares in the 4 small ones you have 8 triangles then in the diamond (tilted square) you have 8 but 4 overlap with the previous
then in the big one you have another 8 big triangles so (4*8)+(4+8)= 44
Hope this helps :).
• Mary, Oregon
•
• Use your crayons, children. Replicate the large square as many times as you need to allow you to color each size of triangle a different color. Unless you are colorblind, you will only be able to color 44 triangles.
• Yr 6 REB, British School Of Kuwait
•
• We all tried really hard at this one! Most of us enjoyed it! Thankyou! Arwa managed to get 44! She got our housepoint today! Thankyou for this starter! We had fun :-).
• SEA, 9A
•
• Samworth Year 9. We think the answer is 44.
•
• We think its 44!!!!!
• Miss Watson's Class, North Yorks
•
• We think the answer is 44. Easy!
• Year 4L, Seoul Foreign British School
•
• We think it is 44!
• Edmead's Barmy Army, Bay House UK
•
• We think there are 44 triangles in total.
• Apinaya, Swindon
•
• 44 I am so sure we checked the answer.
• 10MEA2015, Wythenshawe
•
• We think its 44 :).
• Maria, California
•
• 40!!!! I just went around and counted them 5 times to make sure.
• Alexis, USA
•
• Well, I got thirty six. I am absolutely sure it is correct as I checked too.
• Hashtag YCIS, Pluto
•
• 44 Triangles is the right answer for this question!!!!!!!
#YCISisawesome, #welikehashtags, #whydidihashtagthis.
• John, Nottingham
•
• Good activity - we got 36.
•

[Transum: Yes, the answers are further down the page for those who have a Transum account.]
• MA, SM
•
• We think George is on the money. We agree with 48 triangles.
•
• We found 34.
• Y10 Isca, Exeter
•
• We found 44.

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For Students:

For All:
user
### Content Structure:
The text passage presents a visual puzzle asking how many triangles are hidden in a pattern. It includes a question, a hint for a strategy, and a series of comments from various individuals and classes, most of whom agree on the answer being 44 triangles. The comments also include some explanations and justifications for their counts.

### Analysis:
TYPE: SINGLE_QUESTION_WITH_ANSWER

1. **Mathematical Question**: The passage contains a clear mathematical question: "How many triangles are hidden in the pattern?"
2. **Answer Provided**: The answer is provided in the comments, with a majority consensus on 44 triangles. The answer is also confirmed at the bottom of the page.

### Explanation:
- **Initial Question**: The passage starts with a question asking how many triangles are hidden in the pattern.
- **Strategy Hint**: It suggests a strategy for counting the triangles: "What strategy might you use to count them all to ensure you don't miss any out?"
- **Comments and Answers**: The comments section includes various individuals and classes providing their counts, with most agreeing on 44 triangles. Some comments provide detailed explanations of how they arrived at their counts, breaking down the triangles by size and position.
- **Confirmation**: The correct answer is