How Many Triangles?
How Many Triangles?
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Keywords: Starter
| Problem Solving
| Shape
16 + (8X2) + (4X2) + (1X2) + (1X2)
16= smalls ones
(8X2) = X2 triangles
(4X2) = X4 triangles
(2X2) = half of the square."
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."
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)"
Some argue that at 48 some are counted twice. I assure you, they are not counted twice. Look again."
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! :)"
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."
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."
I did just notice the overlap of the ones with 2 small triangles. Sorry!"
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?"
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."
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."
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."
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. ."
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."
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."
I counted 44."
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."
Good puzzle, made me think a bit."
Excellent, I would like to see more like this
Good, achieved the results I required
Satisfactory
Didn't really capture the interest of the students
Not for me! I wouldn't use this type of activity.
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