I don’t think I did. Using the original geometry except with areas 1 having 14 breads side-by-side and area 3 having 14 breads per layer (12 side-by-side and 2 extra to eat up the remaining 2x12 space), we come to a total of
2x3x14+2x3x12+14x3=198
which requires only the 12 extra breads I accounted for with my updated geometry.
I am so confused right now
I can draw some helpful pictures describing my geometry later when I have a chance to do so, unless Vyse can come up with a problem in my solution.
Ah, that’s where you got that number from. In that case, congrats, it works. The intended solution was an approximation of a sphere, having 3 layers of 3 by 3, then 13 layers of 3 by 4, and then again 3 layers of 3 by 3. Viewed from the side, where the 3 by 3 layer and 3 by 4 layer are touching:
3 by 3 layer: - - -
3 by 4 layer: - - - -
@midsummer I’d ask you to make those helpful pictures so that everyone can follow your solution as well.
What about triangle method? I think it works too
Regarding the triangle method, it seems similar in principle to what I was doing, except I used a square instead of a triangle. My solution just barely reached 210. I think that a triangle shape would not use space as effectively as a square shape (at least the area in the middle can’t be filled as effectively), so I think it’d probably fall a bit short of 210, but without actually running the calculations, one never knows.
If there are 21 breads stacked side by side in each side of the triangle, going from the bottom right corner of a side to the upper left corner takes a distance of sqrt(19^2+7^2) > 21.
Thus, there are some breads in the lowest layer that are more than 21 cm away from some breads in the highest layer.
You dont have to go to the uppercorner, since they are standing on each other the closest point would be the second triandle’s peak.
There are three layers. Every bread in the lowest layer must be able to reach every bread in the upmost layer in less than 21 cm. Either I’ve understood something wrong about your geometry, or it just doesn’t seem to work out.
Let’s take one entire side of the triangle, that is, a stack of 21x3 gingerbreads, if I’ve understood correctly. How far is the distance between the bread in the bottom right and the bread in the top left of this stack?
Actually, nevermind, it seems like I calculated stuff wrong since there’s only a distance of 19 breads in the x-direction. To my surprise, sqrt(19^2+7^2) is actually less than 21. So, it seems like it checks out.
…However, if I don’t confine things into a single side and instead calculate the length of the line in the picture above, I arrive at roughly 22.4, which is more than 21.
I already calculated it above, apothem is around 18,19 ((a*sqrt 3)/2)
then just use pythagorean and its sqrt out of 379,9
I’m not that familiar with English terminology, but based on a quick Google search, it doesn’t seem like the apothem is what should be used in calculating the length of that line.
Shouldn’t the length of x in that picture where you’ve calculated it be
sqrt(7^2+sqrt(21^2+(21/2)^2)^2) = 24.5
where sqrt(21^2+(21/2)^2) is the length of the third side of that red triangle.
Oh yeah, I screwed up with my pythagorean, the length of that one side of the red triangle isn’t sqrt(21^2+(21/2)^2), it’s sqrt(21^2-(21/2)^2). My bad. I didn’t remember the direct formula for the height of an equilateral triangle so I instead used pythagorean but I made a stupid mistake.
Yeah, it seems like your solution works just fine, and you posted it before me, so congratulations.
I like your idea too 
But why was it 6 difficulty?
Helpful picture, as promised. The geometry consists of three layers that are shown from above in the picture. There’s an additional side view to show how one pair of walls works since it is a bit different in layer 2, unlike all other stacks which behave exactly the same in every layer.
The red squares show the edges of the 12.12x12.12x12.12 cube inside which every bread must at least partially be. If all 210 breads touch that cube, then it is impossible for any two breads to be further than 21 cm away from each other.
Wow, that’s so neat, unlike my drawings 
Let’s see how long you are going to take to add all that up.
