SUGGESTED ANSWERS TO PUZZLE





TINA
Ok let's see if this is the right answer. Both of the triangle"s are the same size in width and length. They both have the same number of squares in each color. The only difference is that the color's are rotated in different places.As I said each color has the same amout of spaces but if you notice the triangle at the top with the yellow boxe's are laid over five green squares and the triangle at the bottom, the yellow squares are only laid over two of the green squares still giving it the same width and length but color's are just in different places. ..............Please tell me if this is correct. Thank's, Tina


JEREMY
it is possible because the leg of the green figure was 3 units and the leg of the orange is only 2 units. so when u put the green on top of the orange the empty space appears. The red triangle and the green triangle's simply changed places keeping the same number of blocks in the entire figure.


SCOTT
RE: PUZZLE....you created a wormhole...sortof...because the two triangle images you see are an optical illusion of sorts. They appear to be identical, but they are not. The larger red triangle...the base angle...is slightly obtuse by a few degrees ie: 91° or whatever. The hypotinous (sp?) is NOT a straight line. The base angle of the smaller dk. green triangle, may or may not be 90°. Most likely slightly less than 90° ie: 89° These angle variations are basically undetectable by the naked eye. In the 1st image, the red triangle is on the bottom left...its bottom left angle is smaller than the bottom left angle of the dk. green triangle, and has a longer 'rise' to it, thereby using MORE space and all the quadrants are filled. In the 2nd image with the dk. green triangle on the bottom left and the red triangle above right, the opposite effect occurs...LESS space used to fill the quadrents. The minute quanity of accrued space as a result of the angle projection is then "repr


STAN
the answer i think is quite simple. the area of the predetermined objects do not equal the sum of the total object (objects 32 total 32 1/2) this is a mathmatical quirk that feel cannot be solved. Your question was why rearranged do they not match up and that was easy like i said if some pointy head college geek can explain this then please educate me


SCOTT
re:puzzle....all things being equal...as 2 houses may have the same area, they can be 'shaped' as diff. as night n day. :‹)


GINNY
I am no good for that puzzle not smart enough but when i looked at it this is what i think i saw same dementions on both the hole resulted just from the placement of the second puzzle the way the triangles were arranged if you can understand what i mean.


NICK
Piece of cake! Picture one shows a green triangle on top, and a red triangle to the left: so in order to fill the space under the green triangle and to the right of the red triangle, you must have the odd shaped boxes completely on top of one another almost in a "square shape". NOW PICTURE TWO shows the red and green triangle to have "Traded spots" leaving the space to fill much lower in height and also longer in width. SO the only way to fill this space with the remaining pieces is to match the 2 remaining odd shaped boxes in such a way that leaves a gap along the bottom. But Observe!!! The space the odd boxes fill is a rectangle, so you could put the odd boxes in there upside down and you will still have "Somewhat filled the spot ; all except for a hole in the puzzle now nearer the center. And that is it. That is the definate explanation of why it has to work that way.


BUDDY
I think the solution is simpler than what we make it.  The puzzle pieces are simply rearranged, once to form a perfect triangle , once not. The sum total of the pieces might contain the same area , but according to the way you arrange them , once they form a triangle and once they do not.


MIKE
The angle of the top triangle is bowed enough to keep the area of the bottom triangle the same. If a straight line was drawn from the bottom left side of the top triangle to the top, the area covered will result in the "extra square".


WARREN
Your top blue/red line has a slight sag in it to accout for the difference


ADAM
The top triangle is made of 4 coloured regions Red triangle green triangle Yellow region and light green region. This gives the illusion of 4 areas all the same size. It is not strictly so. There are 4 regions of the same SIZE. But in fact only 2 regions the same AREA. They are the Red and Dark Green tiangle areas. The third region is an area comprising the yellow and light green six sided regions combined making a total of 5 x 3 squares making an area size of 15 squares. The second arrangement has the same regions Red and Dark green triangles (exactly the same as the first triangle. the remaining region however is 2 x 8 squares making its AREA = to 16 squares. this anomaly comes from the fact that the two red and dark green triangles are pythagorean. the length of the hypotenuse (or sloped side) is equal to the total square root of (the sum of the mathematical square's of the length of the other two sides). So the trick is to look at the puzzle as three area's the red triangle the dark green triangle but combine the remaning total area as one space..use the pythagorean formula to calculate the remaining lenthsa when you swap the position of the two smaller triangles.. (God this sounds so long winded...in simple terms swapping the red and green triangles along the hypotenuse introduces an non linear change in the area beneath the triangles because the Square root part of Pythagoras's formula introduces this non linearity. For example of this non linearity..remember 2 squared is 4 but 3 squared is 9.. :o)


ROGER
I don't know much bout math but it's not hard to see why the hole is there. the red triangle is 8 blocks long 3 high the green triangle is 5 blocks long and 2 high. Switching these two pieces as you did (forward point to rear upper piont) ... you now have changed the amount of area for the yellow and green figures to fill. although you have the same outer dementions the inner dementions cannot stay the same. Mathmatically I don't know.


BOB
One of my former students asked me to take a look at your triangle puzzle. The "trick" is that the two smaller triangles are not similar. I'm assuming that all sides that appear to be horizontal are indeed at right angles to those sides that appear to be vertical, and that every side begins and ends exactly on a regularly spaced coordinate grid point. The hypotenuse of the dark blue-green triangle is actually slightly "steeper" than that of the red one. When it is on top, as in the first diagram, the slanted side of the whole "triangle' actually bends in a bit where the two small triangles meet. The large figure is not a triangle at all, but has four sides. The lower figure bends out a bit where the two small triangles meet, so it is actually a four-sided figure as well. This outward bend is enough to add the area of the extra square.


STEPHEN
Your multicolored, conglomerate triangle is not a triangle.  This is easily deduced by the fact that the red and the green triangles' hypoteneuses have different slopes.  The hypoteneuse of the red triangle has a slope of 3/8, while the hypoteneuse of the green triangle has a slope of 2/5.  Since they have different slopes, they are not on the same line and therefore cannot combine to make a hypoteneuse of the conglomerate triangle.  The area in the gap of the second diagram can be accounted for by the rearrangement, since neither are actually triangles.  (start at the bottom right hand corner of the green boot shaped piece, move -5 in the x direction and +3 in the y direction.  do this on both diagrams.  You will find that you are not on the hypoteneuse in the second diagram, because of the differing slope.  you are actually inside of the conglomerate triangle.  there is your missing area!)