------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped
On 25/10/2022 05:23, henh...@gmail.com wrote:
------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped
What do you count as separate combinations? Are rotated/translated combinations distinct?
Mike.
On Tuesday, October 25, 2022 at 6:24:10 AM UTC-7, Mike Terry wrote:
On 25/10/2022 05:23, henh...@gmail.com wrote:
------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped
What do you count as separate combinations? Are rotated/translated combinations distinct?
Mike.
we can assume that each stamp in the (X times Y) sheet has a different picture.
for me ... i'm most curious if there's an answer that comes out simply as
e.g.
(x-1)C(z-1) times (y-1)C(z-1)
On 25/10/2022 17:35, henh...@gmail.com wrote:
On Tuesday, October 25, 2022 at 6:24:10 AM UTC-7, Mike Terry wrote:
On 25/10/2022 05:23, henh...@gmail.com wrote:
------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped
What do you count as separate combinations? Are rotated/translated combinations distinct?
Mike.
we can assume that each stamp in the (X times Y) sheet has a different picture.
for me ... i'm most curious if there's an answer that comes out simply as e.g.Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.
(x-1)C(z-1) times (y-1)C(z-1)
All the answers are like that, but we have to appropriately account for rotations/reflections, and
the different shapes that are included in the set to be counted.
So... for 4-stamp combinations: (with a 4x3 sheet)
h v rr Tot
xxxx : 1 x 3 x 1 = 3
x : 4 x 0 x 1 = 0
x
x
x
xxx : 2 x 2 x 4 = 16
x
xx : 3 x 1 x 4 = 12
x
x
xxx : 2 x 2 x 2 = 8
x
x : 3 x 1 x 2 = 6
xx
x
xx : 3 x 2 x 1 = 6
xx
xx : 2 x 2 x 2 = 8
xx
x : 3 x 1 x 2 = 6
xx
x
Total: 65
[ columns: h = horizontal placements, v = vertical, rr = rotations/reflections ]
If you're saying you would like a simple formula to get the final number 65, there is not going to
be any such 'simple' formula.
On Friday, October 28, 2022 at 12:43:33 PM UTC-7, Mike Terry wrote:
On 25/10/2022 17:35, henh...@gmail.com wrote:
On Tuesday, October 25, 2022 at 6:24:10 AM UTC-7, Mike Terry wrote:Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.
On 25/10/2022 05:23, henh...@gmail.com wrote:
------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped >>>>>
What do you count as separate combinations? Are rotated/translated combinations distinct?
Mike.
we can assume that each stamp in the (X times Y) sheet has a different picture.
for me ... i'm most curious if there's an answer that comes out simply as >>> e.g.
(x-1)C(z-1) times (y-1)C(z-1)
All the answers are like that, but we have to appropriately account for rotations/reflections, and
the different shapes that are included in the set to be counted.
So... for 4-stamp combinations: (with a 4x3 sheet)
h v rr Tot
xxxx : 1 x 3 x 1 = 3
x : 4 x 0 x 1 = 0
x
x
x
xxx : 2 x 2 x 4 = 16
x
xx : 3 x 1 x 4 = 12
x
x
xxx : 2 x 2 x 2 = 8
x
x : 3 x 1 x 2 = 6
xx
x
xx : 3 x 2 x 1 = 6
xx
xx : 2 x 2 x 2 = 8
xx
x : 3 x 1 x 2 = 6
xx
x
Total: 65
i got 65 too.
[ columns: h = horizontal placements, v = vertical, rr = rotations/reflections ]
If you're saying you would like a simple formula to get the final number 65, there is not going to
be any such 'simple' formula.
i'd love to see a simple formula for this (or similar) problem.
__________________________
from a 12-square array ( 4 x 3 sheet ) How many 4-stamp combinations can you create ?
Assuming that the ans. here is 65...
from a 1200-square array ( 40 x 30 sheet ) How many 4-stamp combinations can you create ?
---------- is it bigger or smaller than 6500 ?
On Friday, October 28, 2022 at 12:43:33 PM UTC-7, Mike Terry wrote:
On 25/10/2022 17:35, henh...@gmail.com wrote:
On Tuesday, October 25, 2022 at 6:24:10 AM UTC-7, Mike Terry wrote:
On 25/10/2022 05:23, henh...@gmail.com wrote:
------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped >>>
What do you count as separate combinations? Are rotated/translated combinations distinct?
Mike.
we can assume that each stamp in the (X times Y) sheet has a different picture.
for me ... i'm most curious if there's an answer that comes out simply as e.g.Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.
(x-1)C(z-1) times (y-1)C(z-1)
All the answers are like that, but we have to appropriately account for rotations/reflections, and
the different shapes that are included in the set to be counted.
So... for 4-stamp combinations: (with a 4x3 sheet)
h v rr Tot
xxxx : 1 x 3 x 1 = 3
x : 4 x 0 x 1 = 0
x
x
x
xxx : 2 x 2 x 4 = 16
x
xx : 3 x 1 x 4 = 12
x
x
xxx : 2 x 2 x 2 = 8
x
x : 3 x 1 x 2 = 6
xx
x
xx : 3 x 2 x 1 = 6
xx
xx : 2 x 2 x 2 = 8
xx
x : 3 x 1 x 2 = 6
xx
x
Total: 65i got 65 too.
[ columns: h = horizontal placements, v = vertical, rr = rotations/reflections ]
If you're saying you would like a simple formula to get the final number 65, there is not going toi'd love to see a simple formula for this (or similar) problem.
be any such 'simple' formula.
__________________________
from a 12-square array ( 4 x 3 sheet ) How many 4-stamp combinations can you create ?
Assuming that the ans. here is 65...
from a 1200-square array ( 40 x 30 sheet ) How many 4-stamp combinations can you create ?
---------- is it bigger or smaller than 6500 ?
On Friday, October 28, 2022 at 4:19:49 PM UTC-4, henh...@gmail.com wrote:
On Friday, October 28, 2022 at 12:43:33 PM UTC-7, Mike Terry wrote:
On 25/10/2022 17:35, henh...@gmail.com wrote:
On Tuesday, October 25, 2022 at 6:24:10 AM UTC-7, Mike Terry wrote:
On 25/10/2022 05:23, henh...@gmail.com wrote:
------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped >>>
What do you count as separate combinations? Are rotated/translated combinations distinct?
Mike.
we can assume that each stamp in the (X times Y) sheet has a different picture.
for me ... i'm most curious if there's an answer that comes out simply asWell, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.
e.g.
(x-1)C(z-1) times (y-1)C(z-1)
All the answers are like that, but we have to appropriately account for rotations/reflections, and
the different shapes that are included in the set to be counted.
So... for 4-stamp combinations: (with a 4x3 sheet)
h v rr Tot
xxxx : 1 x 3 x 1 = 3
x : 4 x 0 x 1 = 0
x
x
x
xxx : 2 x 2 x 4 = 16
x
xx : 3 x 1 x 4 = 12
x
x
xxx : 2 x 2 x 2 = 8
x
x : 3 x 1 x 2 = 6
xx
x
xx : 3 x 2 x 1 = 6
xx
xx : 2 x 2 x 2 = 8
xx
x : 3 x 1 x 2 = 6
xx
x
Total: 65i got 65 too.
[ columns: h = horizontal placements, v = vertical, rr = rotations/reflections ]
If you're saying you would like a simple formula to get the final number 65, there is not going toi'd love to see a simple formula for this (or similar) problem.
be any such 'simple' formula.
__________________________
from a 12-square array ( 4 x 3 sheet ) How many 4-stamp combinations can you create ?
Assuming that the ans. here is 65...
from a 1200-square array ( 40 x 30 sheet ) How many 4-stamp combinations can you create ?
---------- is it bigger or smaller than 6500 ?for n x m sheet with n, m >= 3:
xxxx (n-3)*m+(m-3)*n
xxx 4*((n-2)*(m-1)+(n-1)*(m-2))
x
xxx 2*((n-2)*(m-1)+(n-1)*(m-2))
x
xx 2*((n-2)*(m-1)+(n-1)*(m-2))
xx
xx (n-1)*(m-1)
xx
Total: 8*((n-2)*(m-1)+(n-1)*(m-2))+(n-3)*m+(m-3)*n+(n-1)*(m-1)
For 40 x 30 this gives 20,873
Please reply to ilanlmayer at gmail dot com
__/\__
\ /
__/\\ //\__ Ilan Mayer
\ /
/__ __\ Toronto, Canada
/__ __\
||
On Saturday, October 29, 2022 at 9:42:09 AM UTC-4, Ilan Mayer wrote:
On Friday, October 28, 2022 at 4:19:49 PM UTC-4, henh...@gmail.com wrote: >>> On Friday, October 28, 2022 at 12:43:33 PM UTC-7, Mike Terry wrote:
for n x m sheet with n, m >= 3:On 25/10/2022 17:35, henh...@gmail.com wrote:i got 65 too.
On Tuesday, October 25, 2022 at 6:24:10 AM UTC-7, Mike Terry wrote: >>>>>> On 25/10/2022 05:23, henh...@gmail.com wrote:Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.
------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped >>>>>>>
What do you count as separate combinations? Are rotated/translated combinations distinct?
Mike.
we can assume that each stamp in the (X times Y) sheet has a different picture.
for me ... i'm most curious if there's an answer that comes out simply as >>>>> e.g.
(x-1)C(z-1) times (y-1)C(z-1)
All the answers are like that, but we have to appropriately account for rotations/reflections, and
the different shapes that are included in the set to be counted.
So... for 4-stamp combinations: (with a 4x3 sheet)
h v rr Tot
xxxx : 1 x 3 x 1 = 3
x : 4 x 0 x 1 = 0
x
x
x
xxx : 2 x 2 x 4 = 16
x
xx : 3 x 1 x 4 = 12
x
x
xxx : 2 x 2 x 2 = 8
x
x : 3 x 1 x 2 = 6
xx
x
xx : 3 x 2 x 1 = 6
xx
xx : 2 x 2 x 2 = 8
xx
x : 3 x 1 x 2 = 6
xx
x
Total: 65
[ columns: h = horizontal placements, v = vertical, rr = rotations/reflections ]i'd love to see a simple formula for this (or similar) problem.
If you're saying you would like a simple formula to get the final number 65, there is not going to
be any such 'simple' formula.
__________________________
from a 12-square array ( 4 x 3 sheet ) How many 4-stamp combinations can you create ?
Assuming that the ans. here is 65...
from a 1200-square array ( 40 x 30 sheet ) How many 4-stamp combinations can you create ?
---------- is it bigger or smaller than 6500 ?
xxxx (n-3)*m+(m-3)*n
xxx 4*((n-2)*(m-1)+(n-1)*(m-2))
x
xxx 2*((n-2)*(m-1)+(n-1)*(m-2))
x
xx 2*((n-2)*(m-1)+(n-1)*(m-2))
xx
xx (n-1)*(m-1)
xx
Total: 8*((n-2)*(m-1)+(n-1)*(m-2))+(n-3)*m+(m-3)*n+(n-1)*(m-1)
This can be simplified into 19*n*m-28*n-28*m+33
For 40 x 30 this gives 20,873
Please reply to ilanlmayer at gmail dot com
__/\__
\ /
__/\\ //\__ Ilan Mayer
\ /
/__ __\ Toronto, Canada
/__ __\
||
On 29/10/2022 18:54, Ilan Mayer wrote:
On Saturday, October 29, 2022 at 9:42:09 AM UTC-4, Ilan Mayer wrote:
On Friday, October 28, 2022 at 4:19:49 PM UTC-4, henh...@gmail.com wrote: >>> On Friday, October 28, 2022 at 12:43:33 PM UTC-7, Mike Terry wrote:
for n x m sheet with n, m >= 3:On 25/10/2022 17:35, henh...@gmail.com wrote:i got 65 too.
On Tuesday, October 25, 2022 at 6:24:10 AM UTC-7, Mike Terry wrote: >>>>>> On 25/10/2022 05:23, henh...@gmail.com wrote:Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.
------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped >>>>>>>
What do you count as separate combinations? Are rotated/translated combinations distinct?
Mike.
we can assume that each stamp in the (X times Y) sheet has a different picture.
for me ... i'm most curious if there's an answer that comes out simply as
e.g.
(x-1)C(z-1) times (y-1)C(z-1)
All the answers are like that, but we have to appropriately account for rotations/reflections, and
the different shapes that are included in the set to be counted.
So... for 4-stamp combinations: (with a 4x3 sheet)
h v rr Tot
xxxx : 1 x 3 x 1 = 3
x : 4 x 0 x 1 = 0
x
x
x
xxx : 2 x 2 x 4 = 16
x
xx : 3 x 1 x 4 = 12
x
x
xxx : 2 x 2 x 2 = 8
x
x : 3 x 1 x 2 = 6
xx
x
xx : 3 x 2 x 1 = 6
xx
xx : 2 x 2 x 2 = 8
xx
x : 3 x 1 x 2 = 6
xx
x
Total: 65
[ columns: h = horizontal placements, v = vertical, rr = rotations/reflections ]i'd love to see a simple formula for this (or similar) problem.
If you're saying you would like a simple formula to get the final number 65, there is not going to
be any such 'simple' formula.
__________________________
from a 12-square array ( 4 x 3 sheet ) How many 4-stamp combinations can you create ?
Assuming that the ans. here is 65...
from a 1200-square array ( 40 x 30 sheet ) How many 4-stamp combinations can you create ?
---------- is it bigger or smaller than 6500 ?
xxxx (n-3)*m+(m-3)*n
xxx 4*((n-2)*(m-1)+(n-1)*(m-2))
x
xxx 2*((n-2)*(m-1)+(n-1)*(m-2))
x
xx 2*((n-2)*(m-1)+(n-1)*(m-2))
xx
xx (n-1)*(m-1)
xx
Total: 8*((n-2)*(m-1)+(n-1)*(m-2))+(n-3)*m+(m-3)*n+(n-1)*(m-1)
This can be simplified into 19*n*m-28*n-28*m+33
Right - a 2nd degree polynomial in m,n.
As m,n increase (together), the 19*m*n term dominates, and the result, divided by n*m (number of
stamps) converges to 19.
19 is the number of essentially different shapes we can make with 4 connected stamps. Mike.
On Saturday, October 29, 2022 at 11:08:48 AM UTC-7, Mike Terry wrote:
On 29/10/2022 18:54, Ilan Mayer wrote:
On Saturday, October 29, 2022 at 9:42:09 AM UTC-4, Ilan Mayer wrote:
On Friday, October 28, 2022 at 4:19:49 PM UTC-4, henh...@gmail.com wrote:
On Friday, October 28, 2022 at 12:43:33 PM UTC-7, Mike Terry wrote: >>>> On 25/10/2022 17:35, henh...@gmail.com wrote:for n x m sheet with n, m >= 3:
i got 65 too.On Tuesday, October 25, 2022 at 6:24:10 AM UTC-7, Mike Terry wrote: >>>>>> On 25/10/2022 05:23, henh...@gmail.com wrote:Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.
------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped
What do you count as separate combinations? Are rotated/translated combinations distinct?
Mike.
we can assume that each stamp in the (X times Y) sheet has a different picture.
for me ... i'm most curious if there's an answer that comes out simply as
e.g.
(x-1)C(z-1) times (y-1)C(z-1)
All the answers are like that, but we have to appropriately account for rotations/reflections, and
the different shapes that are included in the set to be counted.
So... for 4-stamp combinations: (with a 4x3 sheet)
h v rr Tot
xxxx : 1 x 3 x 1 = 3
x : 4 x 0 x 1 = 0
x
x
x
xxx : 2 x 2 x 4 = 16
x
xx : 3 x 1 x 4 = 12
x
x
xxx : 2 x 2 x 2 = 8
x
x : 3 x 1 x 2 = 6
xx
x
xx : 3 x 2 x 1 = 6
xx
xx : 2 x 2 x 2 = 8
xx
x : 3 x 1 x 2 = 6
xx
x
Total: 65
[ columns: h = horizontal placements, v = vertical, rr = rotations/reflections ]i'd love to see a simple formula for this (or similar) problem.
If you're saying you would like a simple formula to get the final number 65, there is not going to
be any such 'simple' formula.
__________________________
from a 12-square array ( 4 x 3 sheet ) How many 4-stamp combinations can you create ?
Assuming that the ans. here is 65...
from a 1200-square array ( 40 x 30 sheet ) How many 4-stamp combinations can you create ?
---------- is it bigger or smaller than 6500 ?
xxxx (n-3)*m+(m-3)*n
xxx 4*((n-2)*(m-1)+(n-1)*(m-2))
x
xxx 2*((n-2)*(m-1)+(n-1)*(m-2))
x
xx 2*((n-2)*(m-1)+(n-1)*(m-2))
xx
xx (n-1)*(m-1)
xx
Total: 8*((n-2)*(m-1)+(n-1)*(m-2))+(n-3)*m+(m-3)*n+(n-1)*(m-1)
so for (40 x 30) sheet... the ans. is 20,873This can be simplified into 19*n*m-28*n-28*m+33
Right - a 2nd degree polynomial in m,n.
As m,n increase (together), the 19*m*n term dominates, and the result, divided by n*m (number of
stamps) converges to 19.
19 is the number of essentially different shapes we can make with 4 connected stamps. Mike.
i'm counting 18 shapes.... maybe there's a good explanation for why it's 19 (and 28)
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
On Saturday, October 29, 2022 at 11:08:48 AM UTC-7, Mike Terry wrote:
On 29/10/2022 18:54, Ilan Mayer wrote:
On Saturday, October 29, 2022 at 9:42:09 AM UTC-4, Ilan Mayer wrote:
On Friday, October 28, 2022 at 4:19:49 PM UTC-4, henh...@gmail.com wrote: >>>>> On Friday, October 28, 2022 at 12:43:33 PM UTC-7, Mike Terry wrote: >>>>>> On 25/10/2022 17:35, henh...@gmail.com wrote:
for n x m sheet with n, m >= 3:i got 65 too.On Tuesday, October 25, 2022 at 6:24:10 AM UTC-7, Mike Terry wrote: >>>>>>>> On 25/10/2022 05:23, henh...@gmail.com wrote:Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.
------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped >>>>>>>>>
What do you count as separate combinations? Are rotated/translated combinations distinct?
Mike.
we can assume that each stamp in the (X times Y) sheet has a different picture.
for me ... i'm most curious if there's an answer that comes out simply as
e.g.
(x-1)C(z-1) times (y-1)C(z-1)
All the answers are like that, but we have to appropriately account for rotations/reflections, and
the different shapes that are included in the set to be counted.
So... for 4-stamp combinations: (with a 4x3 sheet)
h v rr Tot
xxxx : 1 x 3 x 1 = 3
x : 4 x 0 x 1 = 0
x
x
x
xxx : 2 x 2 x 4 = 16
x
xx : 3 x 1 x 4 = 12
x
x
xxx : 2 x 2 x 2 = 8
x
x : 3 x 1 x 2 = 6
xx
x
xx : 3 x 2 x 1 = 6
xx
xx : 2 x 2 x 2 = 8
xx
x : 3 x 1 x 2 = 6
xx
x
Total: 65
[ columns: h = horizontal placements, v = vertical, rr = rotations/reflections ]i'd love to see a simple formula for this (or similar) problem.
If you're saying you would like a simple formula to get the final number 65, there is not going to
be any such 'simple' formula.
__________________________
from a 12-square array ( 4 x 3 sheet ) How many 4-stamp combinations can you create ?
Assuming that the ans. here is 65...
from a 1200-square array ( 40 x 30 sheet ) How many 4-stamp combinations can you create ?
---------- is it bigger or smaller than 6500 ?
xxxx (n-3)*m+(m-3)*n
xxx 4*((n-2)*(m-1)+(n-1)*(m-2))
x
xxx 2*((n-2)*(m-1)+(n-1)*(m-2))
x
xx 2*((n-2)*(m-1)+(n-1)*(m-2))
xx
xx (n-1)*(m-1)
xx
Total: 8*((n-2)*(m-1)+(n-1)*(m-2))+(n-3)*m+(m-3)*n+(n-1)*(m-1)
This can be simplified into 19*n*m-28*n-28*m+33
so for (40 x 30) sheet... the ans. is 20,873
Right - a 2nd degree polynomial in m,n.
As m,n increase (together), the 19*m*n term dominates, and the result, divided by n*m (number of
stamps) converges to 19.
19 is the number of essentially different shapes we can make with 4 connected stamps. Mike.
i'm counting 18 shapes.... maybe there's a good explanation for why it's 19 (and 28)
On 30/10/2022 00:19, henh...@gmail.com wrote:
On Saturday, October 29, 2022 at 11:08:48 AM UTC-7, Mike Terry wrote:
On 29/10/2022 18:54, Ilan Mayer wrote:
On Saturday, October 29, 2022 at 9:42:09 AM UTC-4, Ilan Mayer wrote:
On Friday, October 28, 2022 at 4:19:49 PM UTC-4, henh...@gmail.com wrote:
On Friday, October 28, 2022 at 12:43:33 PM UTC-7, Mike Terry wrote: >>>>>> On 25/10/2022 17:35, henh...@gmail.com wrote:for n x m sheet with n, m >= 3:
i got 65 too.On Tuesday, October 25, 2022 at 6:24:10 AM UTC-7, Mike Terry wrote: >>>>>>>> On 25/10/2022 05:23, henh...@gmail.com wrote:Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.
------ pls wait 3+ days (Longer if you find it easy or trivial) before posting answers or hints to the following.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
Each of the 2,3,4 stamps must be connected by an edge
for 3-stamp combinations, they'd be horizontal, vertical, or L-shaped
What do you count as separate combinations? Are rotated/translated combinations distinct?
Mike.
we can assume that each stamp in the (X times Y) sheet has a different picture.
for me ... i'm most curious if there's an answer that comes out simply as
e.g.
(x-1)C(z-1) times (y-1)C(z-1)
All the answers are like that, but we have to appropriately account for rotations/reflections, and
the different shapes that are included in the set to be counted. >>>>>>
So... for 4-stamp combinations: (with a 4x3 sheet)
h v rr Tot
xxxx : 1 x 3 x 1 = 3
x : 4 x 0 x 1 = 0
x
x
x
xxx : 2 x 2 x 4 = 16
x
xx : 3 x 1 x 4 = 12
x
x
xxx : 2 x 2 x 2 = 8
x
x : 3 x 1 x 2 = 6
xx
x
xx : 3 x 2 x 1 = 6
xx
xx : 2 x 2 x 2 = 8
xx
x : 3 x 1 x 2 = 6
xx
x
Total: 65
[ columns: h = horizontal placements, v = vertical, rr = rotations/reflections ]i'd love to see a simple formula for this (or similar) problem.
If you're saying you would like a simple formula to get the final number 65, there is not going to
be any such 'simple' formula.
__________________________
from a 12-square array ( 4 x 3 sheet ) How many 4-stamp combinations can you create ?
Assuming that the ans. here is 65...
from a 1200-square array ( 40 x 30 sheet ) How many 4-stamp combinations can you create ?
---------- is it bigger or smaller than 6500 ?
xxxx (n-3)*m+(m-3)*n
xxx 4*((n-2)*(m-1)+(n-1)*(m-2))
x
xxx 2*((n-2)*(m-1)+(n-1)*(m-2))
x
xx 2*((n-2)*(m-1)+(n-1)*(m-2))
xx
xx (n-1)*(m-1)
xx
Total: 8*((n-2)*(m-1)+(n-1)*(m-2))+(n-3)*m+(m-3)*n+(n-1)*(m-1)
This can be simplified into 19*n*m-28*n-28*m+33
so for (40 x 30) sheet... the ans. is 20,873
Right - a 2nd degree polynomial in m,n.
As m,n increase (together), the 19*m*n term dominates, and the result, divided by n*m (number of
stamps) converges to 19.
19 is the number of essentially different shapes we can make with 4 connected stamps. Mike.
i'm counting 18 shapes.... maybe there's a good explanation for why it's 19 (and 28)It might be a mistake on my part - 19 comes from counting all the rr column values in my original table.
Here's the table again, but I've added the rotated/reflected shapes
h v rr Tot
--------------------------------
xxxx : 1 x 3 x 1 = 3
--------------------------------
x : 4 x 0 x 1 = 0
x
x
x
--------------------------------
xxx : 2 x 2 x 4 = 16
x
x
xxx
xxx
x
x
xxx
--------------------------------
xx : 3 x 1 x 4 = 12
x
x
xx
x
x
x
x
xx
x
x
xx
--------------------------------
xxx : 2 x 2 x 2 = 8
x
x
xxx
--------------------------------
x : 3 x 1 x 2 = 6
xx
x
x
xx
x
--------------------------------
xx : 3 x 2 x 1 = 6
xx
--------------------------------
xx : 2 x 2 x 2 = 8
xx
xx
xx
--------------------------------
x : 3 x 1 x 2 = 6
xx
x
x
xx
x
So... that's 19 shapes! You must have missed one?
Mike.
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
henh...@gmail.com presented the following explanation :
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
78
Ammammata explained on 02/11/2022 :
henh...@gmail.com presented the following explanation :
from a 12-square array ( 4 x 3 sheet )
How many 2-stamp combinations can you create ?
How many 3-stamp combinations can you create ?
How many 4-stamp combinations can you create ?
78ops...
2-stamp 17
3-stamp 22
4-stamp 39
2-stamp 17
3-stamp 22
4-stamp 39
i 'm getting... ( 17, 34, 65 )
2-stamp 17
3-stamp 22
4-stamp 39
i 'm getting... ( 17, 34, 65 )
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