------ 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

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

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.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

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)



___________________
Rereading Proust produces a state of mental inertia (6)

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

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.
(x-1)C(z-1) times (y-1)C(z-1)

Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.

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. For a single shape, the formula is simple, but making different
shapes is not simple...

If you want to vary just the width/height of the sheet, then the number 65 above varies in the
obvious way, as the h and v columns adjust, so we could get a simple formula for this (a polynomial
of degree 2 in W,H), at least for W,H >= 4. (For smaller W,H, some shapes may not fit at all,
complicating things.)

Mike.
ps. there may be silly mistakes in above calculation!

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

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.
(x-1)C(z-1) times (y-1)C(z-1)

Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.

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 ?

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 28/10/2022 21:19, 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 as >>> e.g.
(x-1)C(z-1) times (y-1)C(z-1)

Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.

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 ?

You can work it out if you like - just adjust all the calculation lines above by upping the h and v
columns appropriately. E.g. for the xxxx line, the new line will be

h v rr Tot

old xxxx : 1 x 3 x 1 = 3
new xxxx : 37 x 30 x 1 = 1110

You have increased the width,height each by a factor of 10. Each such scaling will increase the Tot
column by a factor of (typically) a bit more than 10, so the new answer will be more than 6500.

Considering the number as a fraction of the total number of stamps, the fraction will tend to a
limit as the width/height of the stamp block increases. The limit is determined by adding all the
rr values in the calculation columns, which essentially gives the total number of distinct "shapes"
[*] that can be made out of the connected stamps. Perhaps you're really more interested in the
problem of how many such shapes can be made.

[*] "distinct shapes" means shapes ignoring duplicates resulting from reflections/rotations/translations.

Mike.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

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 as e.g.
(x-1)C(z-1) times (y-1)C(z-1)

Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.

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 ?

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
/__ __\
||

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

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:

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)

Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.

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 ?

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)

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
/__ __\
||

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

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:

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)

Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.

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 ?

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)

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.

For 40 x 30 this gives 20,873

Please reply to ilanlmayer at gmail dot com

__/\__
\ /
__/\\ //\__ Ilan Mayer
\ /
/__ __\ Toronto, Canada
/__ __\
||

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

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:

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)

Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.

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 ?

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)

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)

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Saturday, October 29, 2022 at 4:19:45 PM UTC-7, 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:

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)

Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.

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 ?

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)

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)

How many 2-stamp combinations can you create ?

2 shapes. CC (Converging Coefficient) is 2

How many 3-stamp combinations can you create ?

6 shapes.

the CC (Converging Coefficient) must be 6 or 7.

--------------- i'd guess 7.

--- SoupGate-Win32 v1.05
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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:

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)

Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.

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 ?

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)

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.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Saturday, October 29, 2022 at 5:46:07 PM UTC-7, Mike Terry wrote:

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:

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)

Well, if we want a 2x2 square, say, and we have a 4x3 sheet, then we can make 3x2 = 6 such squares.

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 ?

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)

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.

i'm (was) counting 18 shapes....

i missed the vertical one because it doesn't fit in the original ( 4 x 3 sheet )

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

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

--
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-=- -=- -=- -=- -=- -=- -=- -=- - -=-
........... [ al lavoro ] ...........

--- SoupGate-Win32 v1.05
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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 ?

78

ops...

2-stamp 17
3-stamp 22
4-stamp 39

--
/-\ /\/\ /\/\ /-\ /\/\ /\/\ /-\ T /-\
-=- -=- -=- -=- -=- -=- -=- -=- - -=-
........... [ al lavoro ] ...........

--- SoupGate-Win32 v1.05
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On Wednesday, November 2, 2022 at 7:02:41 AM UTC-7, Ammammata wrote:

Ammammata explained on 02/11/2022 :

henh...@gmail.com presented the following explanation :

from a 12-square array ( 4 x 3 sheet )

i wonder where this convention of (W x H) --- [Width first] comes from

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

ops...

2-stamp 17
3-stamp 22
4-stamp 39

i 'm getting... ( 17, 34, 65 )


one more variation would be... (the rest of the sheet stays in One Piece)

--- SoupGate-Win32 v1.05
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henh...@gmail.com expressed precisely :

2-stamp 17
3-stamp 22
4-stamp 39

i 'm getting... ( 17, 34, 65 )

maybe I didn't get the picture :)

--
/-\ /\/\ /\/\ /-\ /\/\ /\/\ /-\ T /-\
-=- -=- -=- -=- -=- -=- -=- -=- - -=-
........... [ al lavoro ] ...........

--- SoupGate-Win32 v1.05
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Il giorno Wed 02 Nov 2022 08:08:55p, *henh...@gmail.com* ha inviato su rec.puzzles il messaggio news:00186d10-c37a-4488-84e9- 94872b24f53dn@googlegroups.com. Vediamo cosa ha scritto:

2-stamp 17
3-stamp 22
4-stamp 39

i 'm getting... ( 17, 34, 65 )

ok, I didn't take as valid those shapes not orthogonally connected,
i.e.

xx
x
x

or

x
xx
x

or

x
xx

etc

--
/-\ /\/\ /\/\ /-\ /\/\ /\/\ /-\ T /-\
-=- -=- -=- -=- -=- -=- -=- -=- - -=-
........... [ al lavoro ] ...........

--- SoupGate-Win32 v1.05
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