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bonanova
Daedalian Member

 Posted: Tue Mar 09, 2010 8:19 am    Post subject: 1 Plates on a table This may not be a chestnut, but it should be. I've seen it a couple places. If you haven't seen it, work it out before looking. 13 plates, but no more than 13, can be placed on a dinner table such that no two of them touch. How many plates does it take to completely cover the table? You may assume the plates are identical perfect circles; the table is a perfect rectangle; the center of every plate lies within the table's perimeter, so that it will not fall; and completely cover means completely obscure the table from view looking from above. The answer is 52. Double the diameter of the plates. The table is [just] covered. If a single point were not covered, another plate could have been centered there. Halve all the dimensions [plates and table]. Place four replicas 2x2 and coalesce them. This works for any number, of course. But saying N plates might lead to a lucky guess of 4N._________________ Vidi, vici, veni.
groza528
No Place Like Home

 Posted: Tue Mar 09, 2010 10:23 am    Post subject: 2 Are the initial 13 plates permitted to hang over the edge?
Quailman
His Postmajesty

Posted: Tue Mar 09, 2010 11:23 am    Post subject: 3

 bonanova wrote: ...the center of every plate lies within the table's perimeter, so that it will not fall...

I'm guessing the answer is yes.
Zag
Tired of his old title

 Posted: Tue Mar 09, 2010 2:28 pm    Post subject: 4 I split out this puzzle while it is under discussion, since I don't want to put a lot of discussion in the chestnuts thread. I don't know about a table that just fits 13 plates, but I think it is pretty clear that the 4N solution is not correct. Consider this arrangement -- it is pretty clear that you can finish covering the table with a lot fewer than 32 plates. Just 10 more will plug up all the holes, so you can cover it with 18 plates
Chuck
Daedalian Member

 Posted: Tue Mar 09, 2010 3:54 pm    Post subject: 5 It could be a very narrow table with the 13 plates in a straight line.
Jack_Ian
Big Endian

 Posted: Tue Mar 09, 2010 4:38 pm    Post subject: 6 If it's rewritten to say "How many plates does it take to guarantee completely covering the table?", then the solution given should work. It should also work for non-rectangular tables. Something like… If the maximum number identical and perfectly circular plates that you can place on a table, such that no two plates touch, is N. Then how many plates would you need to ensure completely obscuring the tabletop when seen from above?
bonanova
Daedalian Member

 Posted: Tue Mar 09, 2010 6:21 pm    Post subject: 7 Yes, the description of the original plates is wrong. Create a hexagonal tiling of the plane. Case 1: dense original plates. Center an inscribing plate on each tile. [Decrease the plate diameter by an infinitessimal amount if you wish, so they do not touch.] Increase the plate diameter so they circumscribe the tiles. The plane is covered. Now shrink the plane so the plates become original size. The area ratio of circum- to in-scribing circles is 4/3. So here a 4/3 density increase suffices. Case 2: sparse original plates. Place an inscribing plate, regularly, on only 1/3 of the hex tiles. [Increase the plate diameter by an infinitessimal amount if you wish.] This is the sparsest layout that does not admit another non-touching plate. Here, by comparison to the first case, a fourfold density increase is needed. Thus for any initial layout, 4/3 x is necessary, while 4 x is sufficient. Further reduction might accrue when going from the plane to a table [end effects.] The OP thus needs to carry the sense that the sparsest original layout may have occurred: "I've just placed 13 plates on a table in such a way that another non-touching plate cannot be added. How many plates are required to ensure the table is completely covered?"_________________ Vidi, vici, veni.
ralphmerridew
Daedalian Member

 Posted: Tue Mar 09, 2010 9:32 pm    Post subject: 8 Does the "sparse" possibility also work on very small numbers of plates? (For example, 1 plate can block a circular table with diameter 1.99, but I think it takes 7 plates to completely cover the table; can that be tweaked to a rectangular table?)
ChickenMarengo
Daedalian Member

 Posted: Wed Mar 10, 2010 7:49 pm    Post subject: 9 It can't be tweaked to a rectangular table. You can prove it like this: Suppose you have a rectangular table with n plates on it, and nowhere to add another plate without touching one already there. Divide the table into n regions, so that each point is in the same region as the nearest centre-of-a-plate. Replace all the plates with ones of double the radius. Each larger plate now covers the region containing its centre. If it didn't, then the region contains a point more than 2r from the nearest centre-of-a-plate, so an extra plate could have been placed there to start with. So between them the larger plates cover the table. Shrink the table and plates down by a factor of 2, and make 4 copies of the table. Assemble these into a rectangle the same size as the original table, and covered by 4n plates of the original size. This works for a rectangle, or any other region with can be divided into 4 congruent regions, each similar to itself.
PuzzleScot
Daedalian Member

 Posted: Mon Mar 15, 2010 4:33 pm    Post subject: 10 I think you may have misstated the original puzzle (since you suspect it should be 'a chestnut'). I believe the original arrangement is that N plates are arranged on a table so that no more plates can be placed without touching another. This gives a more mathematically sound answer, so I'll let you hve a go at this version before publishing the answer.
bonanova
Daedalian Member

Posted: Sat Mar 20, 2010 9:31 am    Post subject: 11

 PuzzleScot wrote: I think you may have misstated the original puzzle (since you suspect it should be 'a chestnut'). I believe the original arrangement is that N plates are arranged on a table so that no more plates can be placed without touching another. This gives a more mathematically sound answer, so I'll let you hve a go at this version before publishing the answer.

How does that wording differ in effect from the end of post #7?

 bonanova wrote: "I've just placed 13 plates on a table in such a way that another non-touching plate cannot be added. How many plates are required to ensure the table is completely covered?"

i.e., why would the solution differ?
_________________
Vidi, vici, veni.
Jack_Ian
Big Endian

 Posted: Sat Mar 20, 2010 9:50 am    Post subject: 12 Remember to include this in your general solution.
PuzzleScot
Daedalian Member

Posted: Sun Mar 21, 2010 2:20 pm    Post subject: 13

 Quote: How does that wording differ in effect from the end of post #7?

That covers what I said - I just didn't see that bit of that thread...
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