A construction problem...
The ancient temples in the kindgom of Pantheia are built with dimensions that are exact multiples of a unitum. Further, the dimensions of each room must be all different. For example, a temple with rooms of dimensions 5x11 and 13x5 is not allowed and neither is one with a square room.
You are the architect and are asked to build a new rectangular temple at the top of the flat topped mountain, Polypheme with atleast 2 rooms. The limited space on the mountain top means that you have to minimize the area of the temple. What is the temple with the smallest area that you can construct under the above constraints?
Explan clearly.
You are the architect and are asked to build a new rectangular temple at the top of the flat topped mountain, Polypheme with atleast 2 rooms. The limited space on the mountain top means that you have to minimize the area of the temple. What is the temple with the smallest area that you can construct under the above constraints?
Explan clearly.
posted by littlecow at 2/24/2006 10:09:00 AM
9 Comments:
You cannot build a perfectly rectangular temple (that is all the space within the rectangle is filled with rooms and no part of the rectangle is empty) with the above specifications. This is because atleast 2 rooms in this hypothetical construction would share atleast one side in common and no 2 sides can have a common dimension. In the simple case, visualize a 2 room temple in the rectangle. Assume the rectangle is ABCD and EF is where the room is divided. The common side in bothg the rooms is EF. Divide one of the rooms in to two and then the common side changes but the condition that they have a common side remains no matter how many divisions one makes
@grenade: good idea using photobucket for image uploads in comments. how do u directly link it to the upload in photobukcet ? also, can't you shrink the size of the center room to 1x2 and then keep adding dimensions to shrink the overall size of the temple ?
Ok, The least area I could get is 153 square unitae. Any better answers?
nice! i had assumed that the structure should have the middle room as the smallest (2x1) and i could get a proof that the least is 153. 117 is way better!
grenade has been playing too much quake 4 in a map which resembles this structure being discussed. i have seen it first hand.
My calculations show that the theoretical lower bound is 110 square units. Here's how...
Let us "assume" that grenade has proved that the general plan must have 5 rooms, at least. Now none of the dimensions of the 5 rooms can be repeated. Given that the minimum dimension is 1, in the ideal situation we will find a layout that has lengths 1,2,3 through 9,10 as the 10 dimensions for the 5 rooms. To minimize the sum of 5 areas generated by non-repetitive dimension allocation, we need to pair {1,10},{2,9}...{5,6} for the 5 rooms.
The total area of such a temple is 110. Now, I don't know if that is achievable. But at least we can limit our search to temples with areas of 110 or more.
If we "assume" that collectively you guys have proved that the temple should be a rectangle, we can eliminate 110 as well because 110 would imply that two rectangles with dimensions that add up to 11 should be adjacent - this won't work because in the allocation scheme above, dimensions that add upto 11 are allocated to the same rectangle!
The minimum length of a side of the temple is 6 because it needs to accommodate the width of 3 rectangles (at a minimum = 1+2+3)
111 and 113 are prime - leave them out
112 = 8*14 or 7*16 has the following problem. I played around, can't find a way to fit the rooms in either layout - I may be wrong.
114 = 6 * 19. Again, can't find any fit in this layout
115 = 23*5. We can eliminate this because we need a minimum dimension of 6 on a side
116 = 29*4 . Same problem
117 = 9*13. Grenade has a solution.
I understand that this doesn't qualify as a solution. But if you can eliminate 112 and 114, the Pantheians should take Grenade's advice.
@sisyphus: Good lines of thought. At this point, it appears that there are only four people who struggle with these problems. So I wonder if having points make any more sense. Moreover, I seem to have lost steam in making these posts. But I will post any good ones I see.
@balaks: your posts are eagerly anticipated, so please continue the good work.
I agree with lenscrafter.
May be it is the predictable fatigue of having to challenge people like Grenade with fresh problems... I think xyfactor is a cool idea.
This blog is like a common pet, which we have all fed, cultivated, groomed and pruned. We can't let it go, not right now at least.
I hope people who follow this blog will join me in refusing to let it die.
I have some suggestions...
a. Why don't you give permissions to a few more people (like Grenade and Libran) to post new questions?
b. And how about perhaps expanding the confines of the problem space to include word+math puzzles, etc.?
I also suggest that you post this comment as a new post and see what response your loyal followers have to these ideas to energize the blog.
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