Garden design problem

Ansette · 2006-04-30 06:55:36

Ok this may be hard to explain..

Imagine a square made up of 1x1 squares, of which for a square of length n, there are n² of these squares..

This is our "garden".

Now.. one of these squares has to be for a shrub..

We're given 3 different types of paving slabs.. first of an L shaped one (e.g. 3 squares, cant really explain other than that), then a 1x3 and finally a 1x4.. we have to work out for what n these can work (having started off with the L's.. the "company" then suggests the next options)

The shrub I must add has to be able to be present in any of the 1x1 squares..

Now, we've proved what n will definitely not work using the following ideas:

The square of side n must fit the pattern (n²-1)/3 ∈ Z+
Now, n can be written as or n=3m+p, where p=0,1 or 2 and m=0,1,2
Thus, n² = 9m² +6mp + p²
And, (n²-1)/3 = (9m² +6mp + p² -1)/3
= 3m² +2mp + (p² -1)/3
The co-efficients for both m² and mp are divisible by 3 so thus whether the whole quadratic is divisible by 3 is dependant on this p² -1 term.
Given the values of p earlier for any n, p²=0,1 or 4. And obviously (p² -1)/3 therefore equals either -⅓, 0 or 1. Thus the pattern will only work for squares that are not divisible by 3.

This would work for both L and 1x3 logically as they are both composed of 3 squares.. and a similiar idea would be used for 1x4..

The thing we want to know is that how can we prove that for any n (other than those discounted above) they will work for the L-shaped blocks (which seemingly we've found simply by trial)..

Also the 1x3 and 1x4 are both worse of in terms of their use.. but how can we show this mathematically.. i presume between the Ls and 1x3s that it's due to the fact that the L has a rotational symmetry order of 1 as opposed to the 1x3s order of 2.. thus meaning it can create more shapes when put with others..but again this is all worded and not particuarly mathematical...

Help me!!

krassi_holmz · 2006-04-30 11:13:20

Let's give some graphics:
garden:

######
######
######
######
######
######

Shrub:

###
#S#
###

L-shape:

#####
#1###
#11##
#####
1x3
#####
#####
#222#
#####
1x4
######
#3333#
######

Last edited by krassi_holmz (2006-04-30 11:20:47)

krassi_holmz · 2006-04-30 11:26:52

Ha, ha, I found it!
Pretty interesting.
You have to note 2 thnigs:
1. n=2 IS a solution:

11
10

2. You can create bigger L-shape using smaller L-shapes:

krassi_holmz · 2006-04-30 11:31:18

here're some pictures:

Last edited by krassi_holmz (2006-04-30 11:34:38)

krassi_holmz · 2006-04-30 11:36:44

...So n=2^i is a solution.
Now we have to prove that !exists another slution.
It's getting late here.
I hope I've helped.
I'll go to sleep.

krassi_holmz · 2006-04-30 11:38:28

(If the picture is too small, copy it to your hard disk and enlarge it with Paint)

Ansette · 2006-04-30 13:31:47

I did accept n=2 as a solution, (the first of a few noticed pros of the l-shaped ones over the others) but I can't say I'd really noticed the ability to make the larger Ls from it..

However even when I copy the image to HD and enlarge it i can't really see anything.. (e.g. not a lack of understanding.. problem with the image!)

krassi_holmz · 2006-04-30 18:53:14

...n=5 IS A SOLUTION!!!

#######
# 22  #
#  2 1#
#33b11#
#3 4  #
#  44 #
#######


#######
#72266#
#77261#
#33b11#
#38455#
#88445#
#######

krassi_holmz · 2006-04-30 19:06:00

5+12k is a solution too.

krassi_holmz · 2006-04-30 19:07:39

if n0 is a solutionq then n0+12k is also a solution!!!

krassi_holmz · 2006-04-30 19:17:50

7 is a solution!

krassi_holmz · 2006-04-30 19:36:15

n=10...

krassi_holmz · 2006-04-30 19:40:16

All n != 3k are solutions!!!

krassi_holmz · 2006-04-30 19:45:49

picture:

Last edited by krassi_holmz (2006-04-30 19:47:13)

krassi_holmz · 2006-04-30 20:01:22

enlarging...

Last edited by krassi_holmz (2006-04-30 20:02:35)

krassi_holmz · 2006-04-30 20:06:59

Much better!!!

Last edited by krassi_holmz (2006-04-30 20:08:02)

Ansette · 2006-04-30 22:32:35

I'm about to really upset here.. but we have to find where the shrub can go per n, not just whether there's one solution.. so far I've come up with these..

http://www.filelodge.com/files/room21/537756/Positions.xls

MathsIsFun · 2006-04-30 23:07:08

This is really interesting stuff, and I love the illustrations!

Are these the rules:
* There must be exactly 1 vacant 1x1 square
* Furthermore, that 1x1 square must be able to be in *any* spot
* The rest of the space must be used up exclusively by a) "L" paving blocks, b) "1x3" blocks c) "1x4" blocks (ie three different case, not in combination)

?

krassi_holmz · 2006-04-30 23:50:01

I agree with n<=5, but n=7 can be inproved (for the Ls)

krassi_holmz · 2006-04-30 23:55:02

n=7:
x0xxx0x
0000000
x0x0x0x
x00x00x
x0x0x0x
0000000
x0xxx0x

Last edited by krassi_holmz (2006-05-01 00:09:15)

krassi_holmz · 2006-05-01 00:10:44

for 1x3:
N=5

Math Is Fun Forum

#1 2006-04-30 06:55:36

Garden design problem

#2 2006-04-30 11:13:20

Re: Garden design problem

#3 2006-04-30 11:26:52

Re: Garden design problem

#4 2006-04-30 11:31:18

Re: Garden design problem

#5 2006-04-30 11:36:44

Re: Garden design problem

#6 2006-04-30 11:38:28

Re: Garden design problem

#7 2006-04-30 13:31:47

Re: Garden design problem

#8 2006-04-30 18:53:14

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#9 2006-04-30 19:06:00

Re: Garden design problem

#10 2006-04-30 19:07:39

Re: Garden design problem

#11 2006-04-30 19:17:50

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#16 2006-04-30 20:06:59

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#17 2006-04-30 22:32:35

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Re: Garden design problem

#19 2006-04-30 23:50:01

Re: Garden design problem

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