I should warn you that this is a tricky one. It came via Facebook and god knows what devil invented it but as all the good puzzles it is carefully retold through generations and social networks.
Two triangles consist of exactly the same shapes. Shaped are rearranged in a different order in the bottom triangle and suddenly... one square disappears. How did this "Bermuda Triangle" swallowed the square?
Fill free to print the pictures and play with them, use a ruler. Your thoughts and suggestions are accepted any time until midnight Eastern Time on Sunday, on our Family Puzzle Marathon. You can also email me your drawings to maria@themathmom.com
11 comments:
Great Puzzle!
Let's look at the area of the individual pieces:
Area(Green) = 1/2*B*h = 1/2*(5)*(2) = 5 units
Area(Yellow) = 1/2*B*h = 1/2*(8)*(3) = 12 units
Area(Red) = (2)*(2) + (1)*(3) = 7 units
Area(Blue) = (2)*(1) + (3)*(2) = 8 units
The total area of the pieces = 32 units
BUT -
The area of the larger triangle = 1/2*B*h = 1/2*(13)*(5) = 32.5 units
So, what's going on?
Neither of the larger triangles are "true" triangles. What appears to be hypotenuse is not a straight line. The top figure is a quadrilateral that is slightly smaller than a 13×5 right triangle and the bottom figure is a quadrilateral that is slightly bigger than a 13×5 right triangle. In the bottom figure, being slightly larger allows the introduction of an empty square.
I you put a straight edge along the "hypotenuse" of the bottom figure, you can see the distortion.
Is it really fair to say "two triangles" consist of exactly the same shapes? They're not truly triangles, right? At least that's my feeling, without actually giving away the answer (or at least what I think is the answer).
Not seen lately and still a fine puzzle.
The hypotenuse of both given triangles, the large ones, is (are) not a straight line, tho very close.
Yes the areas of all the segments are the same, but they do not add up to the area of the whole triangle. The large triangles are not triangles, neither one of them.
The slope of the green triangle is barely steeper than the slope of the yellow.
good problem! I had to print it out & look closely to see an answer. My answer is that the even though the yellow and green "triangles" look like they are identical and have the same height and length in both the top and bottom figures, they are actually not 100% identical because the long sides of the "triangles", the hypotenuses, are not straight in either picture. In the top picture, the long side is concave, and in the bottom picture, the long side is convex, and the difference is enough to create the "extra" square.
-TracyZ
One other comment. The sentence "Two triangles consist of exactly the same shapes" is somewhat misleading because as shown in the problem's solution, the shapes are not exactly the same.
-TracyZ
I'm just going to post a comment or two without a diagram. I can't get my ancient corel draw to make a believable diagram.
The first comment is this. The slope of the yellow triangle is 3/8 squares which is 0.375. The slope of the green triangle is 2/5 which is 0.400. Now there doesn't seem to be that much of a difference, but the result is a "line" that does not look like a straight line when magnified. A straight line would give two slopes that are exactly the same and not a "knee" joint as suggested by the difference in slopes. If corel is correct, a straight line would give a small gap between the two lines making up the knee.
The second comment is this: Suppose we figure out the area of the top diagram. And then figure out the area of all the constituent parts.
Triangular Shape
=============
The base of the large right triangle = 13 units.
The height of the large right triangle = 5 units.
Area = 13 * 5/2 = 32.5
Now take all the parts (thank you for coloring them).
Yellow = 8 * 3/2 = 12
Green = 2 * 5 /2 = 5
Blue = 8 Units (just count them)
Red = 7
The total here is 32. There is a half unit missing and I think it is above the "hypotenuse" of the large triangle.
I'm guessing here, but I think that in the second diagram, the knee is much more pronounced and it is bent outwards eating up another 1/2 square.
That's my guess anyway.
OK, I guess I'm stumped but I did see some curiosities that may be part of an explanation. I figured the area of a 13 x 5 triangle to be 32.5 (1/2 b x h). The combined area of the triangles and shapes in the top triangle are 12 (yellow) +5 (green) +15 (red+blue) = 32, not 32.5. And the bottom triangle is 12 + 5 + 16 (red+blue+1) = 33, which is consistent with the top triangle, and intuitively makes sense to me based on the change in shape of the red/blue area, but does not explain why it is not 32.5!
I've been thinking more about this puzzle. I was thinking back (way back!) to HS Geometry and remembered an important lesson. We can't be certain that a picture tells the truth, hence the need for rigorous proofs. I thought I would see if the angles of the green and yellow triangles were the same so I figured the angles from opposite/adjacent and found through a great calculator website that the angle opposite the right angle of the yellow triangle is 20.56 and the angle opposite the green right angle is 21.8. This led me to believe that the hypotenuse of the large triangle is not straight so it is actually not a triangle and the two shapes (triangles) are different. I thought I was onto something and then I realized do we really know that the triangles are right triangles??
I just want to make sure my answer is clear. The top diagram has a "hypotenuse" that is concave inward hiding 1/2 a square. The second or lower diagram has a "hypotenuse" that is convex outward that spreads 1 full square between it and the actual hypotenuse.
Area of a triangle with height and base of 5 and 13 is (13*5)/2 = 32.5
Area of the individual shapes within the big triangle are:
Yellow triangle (8*3)/2 = 12
Green triangle (5*2)/2 = 5
Blue shape = 8 squares
Red shape = 7 squares
Adding up area of all the individual shapes we get 32 squares.
A triangle with a height and base of 5 and 13 cannot be made with only 32 squares, an extra half square would be needed to make a real triangle.
The triangle that seems to be formed with the four shapes is not a real triangle.
Also the hypotenuse of both the triangles do not line up properly. There appears to be a depression where the two triangles meet in the first image (accounting for the missing half square). And also there seems to be an elevation where the two meet in the second image resulting in the addition of an extra half square.
Lulu
You are unbelievable! This is quite a hard puzzle and most of you slowly figured out the answer by yourself without any prior knowledge. Moreover, I confused you by calling this shapes the "triangles". Sorry! My neighbor across the street runs a private investigation firm and I am going to recommend you all as excellent analysts.
A big fat puzzle reward for everyone who dared. Please never hesitate to write your thoughts even when you don't know the exact answer.
To verify that these shapes are indeed not proper triangles, print out the images and use a ruler to connect the hypotenuse points. See that green and yellow triangles end a bit below and a bit above this line.
see you next week!
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