22 replies to this topic

[bonanova]

Spoiler for

orchard.jpg 13t20r?

 

OK that counts as over-achievement I guess.
It's not the fewest trees that can make eighteen rows, but it probably is the fewest that can make twenty.

Honorable mention!

[bonanova]

So the answer is 12 trees making 18 rows.
Can you draw the layout?

[augustinus]

Spoiler for maybe this?

X  B C  X
D E F G
H I J K
X M N X
4 horizontal
4 vertical
4 outer diagonal parts (BD, CG,HM, NK)
6 inner diagonal parts (BFK,EJ, DIN,CEH,FI, GJM)
X's are empty spots

Edited by augustinus, 26 February 2013 - 11:07 AM.

[bonanova]

On the right track.
We need at least three trees per row.

[tammie]

i thought it was 19 rows. 18 of three and one of more right? total using 12 trees? still don`t have the layout but b4 i spend the time trying i want to make sure i have the right information

So the answer is 12 trees making 18 rows.
Can you draw the layout?

[bonanova]

Hi tammie, and welcome, if no one else has said it, to the Den.

 

Yes, you have it right.

 

There are 12 trees.

 

Eighteen rows have 3 and only 3 trees in them.

There is a "bonus" row, if you want to call it that, and it has 4 trees.

[bonanova]

Shamelessly bumping this puzzle. It's worth the effort to solve, IMO.

 

12 trees making 18 rows of 3 trees each.

[Prime]

Spoiler for theoretical limit

Assuming rows are straight lines, no two different rows may share two trees. Since a row consists of at least 3 trees, there are ₃C₂ = 3 combinations of tree pairs in each row.
Thus for 19 rows we need at least 19*3 = 57 pairs of trees. The minimum number that allows for that many is  12 trees. (₁₂C₂ = 66.)
A tree shares a row with other two trees. It cannot share another row with any of those two trees. If maximum rows in which a tree is a participant is x and total number of trees is N,

then 2x + 1 = N. Solving for N=12, x=5.5.

Since no fractional trees are allowed, each tree can participate in at the most 5 rows. 12*5 = 60. However, since each row has 3 trees, 60/3 = 20.
And so it appears a gardener could construct 20 rows with 12 trees.

Now all gardener has left to do is to use these limits as a guide for planting his garden.

[bonanova]

Spoiler for Small clue

The orchard occupied a rectangular plot of land.

To permit optimal sun exposure and air circulation, seven of the trees were planted along its boundary.

The other five were scattered in its interior.

[plasmid]

I've been stuck on this one. Maybe it would help to know whether or not the following figure is a subset of the final answer, since I keep coming back to it and it seems to at least be a local minimum (or whatever the appropriate term would be)

Spoiler for