Brainteaser/Puzzle Corner (Part 5)

hey quanta is it a battle or a war?

 

trojan horse

 

Emm thats a good one eon, never though of that I'd say your right. What made you think of that?

 

congratulations Eon the answer is Trojan Horse

 

this part that I though about it was the very begining when it said "a gift to the gods" since the trojan horse was a seen as a gift to the trojan gods from defeat

this fits perfectly since the horse was a disguise and was very brilliant in its idealism even to this day.

during the night, the troops left the horse and struck down the gates, allowing the greek army to enter the city and destroy it

 

[img width=300 height=260]http://i56.photobucket.com/albums/g177/clouded_fate/other/triangle.png[/img]
lol wut?

for 300, i may increase it in a day or two
no cheating

 

What are we supposed to be solving for?

 

Im assuming we have to solve where that square came from?

I'll be going to play around with that picture in paint now

 

lets see...

The yellow shape is moved down and left to fit next to the greean shape. however they are not a perfect fit and leave an open gap.

The red and blue triangles are switched becase then the two other shapes were moved, they were not a perfect fit.

The result is a shape with the same dimensions, but a small gap being created from the rearangement of pieces.

 

The task wasn't to tell the story of the problem
The confusing bit is that both big triangles have the same dimensions and consist of the same shapes yet their areas are not the same. I remember seeing this task before but I can't seem to remember the explanation for it.

 

I have a guess: the yellow and the green shapes together occupy 15 squares no matter how you place them (in some formations there might be for example 2 squares missing but protruding somewhere else). By rearranging the blue and the red shapes we get a 2x8 space in the bottom-left instead of the 3x5. Since 16 is greater than 15 by 1, there must be 1 missing square somewhere (it could also be in the middle if you flip the green and yellow shapes vertically)

 

Sorry to triple post but others have done it for the sake of an unedited post for the solution.
The two big triangles are not the same, they just look very similar. The blue small triangle has a steepness of 2.5 while the red has a steepness of 2.66' and I've marked where we lose half a square because of swapping them (which actually doubles to 1 whole square if we calculate the area by the formula (base*altitude)/2 and not by adding the areas of the small shapes - this latter method gives the accurate area which will be the same as the first triangle's)
[img width=300 height=260]http://img251.imageshack.us/img251/7886/triangle2hf3.png[/img]
So in conclusion, the missing square is present in the first picture as well, it's just not so noticable.

 

Great thinking Kuvasz!

 

what? explain.

 

I'll try

I said the red and blue triangles have different 'steepnesses'. The red has a 8/3 (you have to go 8/3 squares to the right to be able to go one whole square up) for its hypotenuse and the blue has 5/2. So the hypotenuse is not a straight line in either of the big triangles. The difference in steepness is very important and I've sketched an exaggerated version of what this differences causes:
[img width=300 height=260]http://img204.imageshack.us/img204/9108/trianglesolutionsz1.png[/img]
The grid is unimportant for now of course. Now as you can see the second big 'triangle' leaves more room for the green and the yellow shapes. In this exaggerated pic it would be even more than 1 square but this doesn't look like a normal right triangle which is the base of this task - to give the impression that the big triangles are perfectly normal while they're not. They're actually quadrilaterals.

So in conclusion: not only does the rectangle where the green and the yellow shapes must fit in change but also the size. The extra space in there results in the 'triangle' to take up more space elsewhere. This is why the areas are correct for both big triangles if you add up the areas of the shapes forming it while if you use the formula for calculating the area of a triangle [(base*altitude)/2] it will not give the correct answer because it is not a triangle but a quadrilateral.

Hope I made sense

 

Fine
Here's the simplyfied version:
[img width=300 height=260]http://img207.imageshack.us/img207/2983/trianglediagramlo1.png[/img]
The green bit shows the difference between the two exaggerated 'triangles'. For the original task, that green rhombus would be extremely thin and would occupy the space of exactly 1 square. In the case of the less saturated 'triangle' the extra 1 square (compared to the more saturated one) would be found on the hypotenuse and since the area is constant (because of the 4 shapes), that 1 square must be taken away from somewhere.
:good:

 

:upset: I should have checked in here earlier I've heard this before. I didn't figure it out until I was shown the that triangles did not have the same angle's.

GJ Kuvasz

 

correct, although i must say that people in this forum have some dire need to overcomplicate every math riddle they see.

 

Math and geometry are not my strong points ;D but bilingual classes sure paid off today ;D

 

Us math people believe in the fine art of overkill