[Guest]

1) What is the minimum number of squares that would be sufficient to create each of the following patterns?

2) Twenty-seven identical cubes are placed together to form the object below. If one of the cubes is displaced, four different shapes are possible: one in which the missing cube is at the corner of the stack, one where it is in the middle of an edge of the stack, one in which it is in the middle of a side of the stack, and one in which it is at the core of the stack. If two of the small cubes are removed rather than one, how many different shapes are possible?

[plainglazed]

[Guest]

Plainglazed, ensure you understand the directions completely. (I.e. SQUARES) Good go anyhow.

[Guest]

Sure

OK, here's how I number the cubes in the faces which are listed from front to back:

Face 1 (front):

1 2 3

4 5 6

7 8 9

Face 2 (middle):

10 11 12

13 14 15

16 17 18

Face 3 (back):

19 20 21

22 23 24

25 26 27

[code]

Here are the equivalence classes in terms of which pair of cubes are missing:

Class # 1 (24 in class -- cumulative=24):

(1,2)(1,4)(1,10)(2,3)(3,6)(3,12)(4,7)(6,9)(7,8)(7,16)(8,9)(9,18)(10,19)(12,21)(16,25)(18,27)(19,20)(19,22)(20,21)(21,24)(22,25)(24,27)(25,26)(26,27)

Class # 2 (12 in class -- cumulative=36):

(1,3)(1,7)(1,19)(3,9)(3,21)(7,9)(7,25)(9,27)(19,21)(19,25)(21,27)(25,27)

Class # 3 (24 in class -- cumulative=60):

(1,5)(1,11)(1,13)(3,5)(3,11)(3,15)(5,7)(5,9)(7,13)(7,17)(9,15)(9,17)(11,19)(11,21)(13,19)(13,25)(15,21)(15,27)(17,25)(17,27)(19,23)(21,23)(23,25)(23,27)

Class # 4 (24 in class -- cumulative=84):

(1,6)(1,16)(1,20)(2,7)(2,21)(3,8)(3,10)(3,24)(4,9)(4,19)(6,27)(7,18)(7,22)(8,25)(9,12)(9,26)(10,25)(12,19)(16,27)(18,21)(19,26)(20,27)(21,22)(24,25)

Class # 5 (24 in class -- cumulative=108):

(1,8)(1,12)(1,22)(2,9)(2,19)(3,4)(3,18)(3,20)(4,25)(6,7)(6,21)(7,10)(7,26)(8,27)(9,16)(9,24)(10,21)(12,27)(16,19)(18,25)(19,24)(20,25)(21,26)(22,27)

Class # 6 (12 in class -- cumulative=120):

(1,9)(1,21)(1,25)(3,7)(3,19)(3,27)(7,19)(7,27)(9,21)(9,25)(19,27)(21,25)

Class # 7 (8 in class -- cumulative=128):

(1,14)(3,14)(7,14)(9,14)(14,19)(14,21)(14,25)(14,27)

Class # 8 (24 in class -- cumulative=152):

(1,15)(1,17)(1,23)(3,13)(3,17)(3,23)(5,19)(5,21)(5,25)(5,27)(7,11)(7,15)(7,23)(9,11)(9,13)(9,23)(11,25)(11,27)(13,21)(13,27)(15,19)(15,25)(17,19)(17,21)

Class # 9 (24 in class -- cumulative=176):

(1,18)(1,24)(1,26)(2,25)(2,27)(3,16)(3,22)(3,26)(4,21)(4,27)(6,19)(6,25)(7,12)(7,20)(7,24)(8,19)(8,21)(9,10)(9,20)(9,22)(10,27)(12,25)(16,21)(18,19)

Class #10 (4 in class -- cumulative=180):

(1,27)(3,25)(7,21)(9,19)

Class #11 (24 in class -- cumulative=204):

(2,4)(2,6)(2,10)(2,12)(4,8)(4,10)(4,16)(6,8)(6,12)(6,18)(8,16)(8,18)(10,20)(10,22)(12,20)(12,24)(16,22)(16,26)(18,24)(18,26)(20,22)(20,24)(22,26)(24,26)

Class #12 (24 in class -- cumulative=228):

(2,5)(2,11)(4,5)(4,13)(5,6)(5,8)(6,15)(8,17)(10,11)(10,13)(11,12)(11,20)(12,15)(13,16)(13,22)(15,18)(15,24)(16,17)(17,18)(17,26)(20,23)(22,23)(23,24)(23,26)

Class #13 (12 in class -- cumulative=240):

(2,8)(2,20)(4,6)(4,22)(6,24)(8,26)(10,12)(10,16)(12,18)(16,18)(20,26)(22,24)

Class #14 (24 in class -- cumulative=264):

(2,13)(2,15)(4,11)(4,17)(5,10)(5,12)(5,16)(5,18)(6,11)(6,17)(8,13)(8,15)(10,23)(11,22)(11,24)(12,23)(13,20)(13,26)(15,20)(15,26)(16,23)(17,22)(17,24)(18,23)

Class #15 (12 in class -- cumulative=276):

(2,14)(4,14)(6,14)(8,14)(10,14)(12,14)(14,16)(14,18)(14,20)(14,22)(14,24)(14,26)

Class #16 (12 in class -- cumulative=288):

(2,16)(2,24)(4,18)(4,20)(6,10)(6,26)(8,12)(8,22)(10,26)(12,22)(16,24)(18,20)

Class #17 (24 in class -- cumulative=312):

(2,17)(2,23)(4,15)(4,23)(5,20)(5,22)(5,24)(5,26)(6,13)(6,23)(8,11)(8,23)(10,15)(10,17)(11,16)(11,18)(11,26)(12,13)(12,17)(13,18)(13,24)(15,16)(15,22)(17,20)

Class #18 (12 in class -- cumulative=324):

(2,18)(2,22)(4,12)(4,26)(6,16)(6,20)(8,10)(8,24)(10,24)(12,26)(16,20)(18,22)

Class #19 (6 in class -- cumulative=330):

(2,26)(4,24)(6,22)(8,20)(10,18)(12,16)

Class #20 (12 in class -- cumulative=342):

(5,11)(5,13)(5,15)(5,17)(11,13)(11,15)(11,23)(13,17)(13,23)(15,17)(15,23)(17,23)

Class #21 (6 in class -- cumulative=348):

(5,14)(11,14)(13,14)(14,15)(14,17)(14,23)

Class #22 (3 in class -- cumulative=351):

(5,23)(11,17)(13,15)

[/spoiler]

Superprismatic, excellent job. Wow.

[plainglazed]

Plainglazed, ensure you understand the directions completely. (I.e. SQUARES) Good go anyhow.

Each object in the spoiler was made from the previous object with the addition of one opaque square (reading left to right, top to bottom). Is that not the objective?

[Guest]

Plainglazed, ensure you understand the directions completely. (I.e. SQUARES) Good go anyhow.

I am impressed by plaingazed answer, he is correct. to clarify the steps here are a listing of each step he illustrates.

1. 4 unit square for full space.

2. 3 unit square lower right covering 6,7,8,10,11,12,14,15,16.

3, 2 unit square covering 2,3,5,6.

4. 2 unit square lower left covering 11,12,15,16.

5. 1 unit square covering 3

6. 1 unit square on 11.

7. 1 unit square on 9.

8. 1 unit square on 1 or 5(same results)

9. square turned 45 degrees in center

overlappingsquares.jpg.

and only shape added is square.

[witzar]

Sure

Thanks. Now I see why I get 20 and you get 22.

You treat two cases as equivalent if there exists rotation that transforms one case into another.

And I allow symmetries as well.

For example (1,6) and (1,8) are equivalent for me but not equivalent for you.

[CaptainEd]

I agree with witzar,

Why are (1,6) and (1,8) different classes (#4, #5)?

Why are (2,16) and (2,18) different classes (#16, #18)?

[Guest]

You're right plainglazed, I misread your answer. Superb job. Care to try the others?

[Guest]

Looking at the gold square in number one, I find several ways to do it with 12. aamph illustrated 1 way in post #13. Since 12 seems relatively easy, the answer must be 10 or 11 will continue to seek a better solution.

[Guest]

Success!!!

1. 5 unit square

2. 2 unit square upper left corner covering 1,2,6,7

3. 2 unit square lower right covering 19,20,24,25

4. 3 unit square in center covering 7,8,9,12,13,14,17,18,19

5. 2 unit square upper right covering 4,5,9,10

6. 2 unit square lower left covering 16,17,21,22

Now a series of 1 unit square

7. covering 10

8. covering 13 (center)

9. covering 17

and 45 degrees squares

10. upper left corner

11. lower right corner

[Guest]

Here is my solution for the 6x6 square. After seing the solution for number 1 with 9 squares I'm wondering if there is a better solution, but this will get it started.

1. 6x6 square

(base)

2. 3x3 1,2,3 7,8,9 13,14,15

3. 3x3 22,23,24 28,29,30 34,35,36

(Top Left)

4. 2x2 1,2 7,8

5. 2x2 2,3 8,9

6. 2x2 8,9 14,15

(Top Right)

7. 2x2 4,5 10,11

8. 1x1 11

(Bottom Left)

9. 2x2 26,27 32,33

10. 1x1 26

(Middle Cover)

11. 3x3 15,16,17 21,22,23 27,28,29

(Bottom Right)

12. 2x2 23,24 29,30

13. 2x2 28,29 34,35

14. 1x1 28

(Diamonds)

15. Diamond 19,20,25,26

16. Diamond 11,12,17,18

17. Diamond 15,16,21,22

This solution works ( I think ) Visually I can see it is possible to start with a 5x5 square anchored

in the bottom right corner. I tried this and it requires the same number of steps with just a few tweaks.

seems like there are two solutions. Someone walk through this and verify it is a bit hard to conceptualize.

[Guest]

Great job on this one.

[Guest]

I'm not sure why you are talking about 258 forms, but the original post says to use the least number of squares.

when i said forms i meant how many shapes u can take by removing two squares... checksuperismatic's post .. he put them into cases but he got to a high total as well..i didn't put them into cases i left them as a total.. btw great job superismatic! guess i missed some:) i think u got them all:D

[Guest]

I agree with witzar,

Why are (1,6) and (1,8) different classes (#4, #5)?

Why are (2,16) and (2,18) different classes (#16, #18)?

i think it's clear that superprismatic(whose name i spelled wrong the first time .. sry ) accepts the symmetrics.. so did i.. if u do not accept symmetrics then there is no diff and you would have to take out alot of answers from his. but i think he got them all (symmetrics included) complements again to superprismatic:D

[superprismatic]

Thanks. Now I see why I get 20 and you get 22.

You treat two cases as equivalent if there exists rotation that transforms one case into another.

And I allow symmetries as well.

For example (1,6) and (1,8) are equivalent for me but not equivalent for you.

I'm a bit confused because I don't know what you mean by symmetries. Would you mind defining symmetries?

[CaptainEd]

Superprismatic, For the classes #4 and #5, imagine a plane cutting through 1,5,9,10,14, 18, 19,23, 27.

The cells (6) and (8) are in symmetric locations wrt cell (1).

Similarly, for classes #16 and #18, imagine a plane cutting through 2,5,8,11,14,17,20,23,26.

The cells (16) and (18) are in symmetric locations wrt cell (2).

So, the issue raised by witzar and me is: shouldn't we merge classes #4 and #5 because of symmetry? and shouldn't we merge classes #16 and #18 as well? These mergers would shrink your 22 classes to our 20 classes.

[witzar]

I'm a bit confused because I don't know what you mean by symmetries. Would you mind defining symmetries?

I should have used term "reflection" instead of "symmetry". Sorry for that.

For me cases are equivalent when there is an isometry of a cube that transforms one case into another.

For you cases are equivalent when there is an orientation preserving (or rotational, which is the same) isometry of a cube that transforms one case into another.

[superprismatic]

I should have used term "reflection" instead of "symmetry". Sorry for that.

For me cases are equivalent when there is an isometry of a cube that transforms one case into another.

For you cases are equivalent when there is an orientation preserving (or rotational, which is the same) isometry of a cube that transforms one case into another.

Reflection doesn't seem natural to me for use in saying that two things are the same. I guess I don't

like the fact that it makes things which are of different handedness the same. It's all a matter of definition

anyway. Nobody can argue what's right and what's wrong here. The OP didn't specify precisely what

"different" meant. Thanks for the clarification.

[superprismatic]

Superprismatic, For the classes #4 and #5, imagine a plane cutting through 1,5,9,10,14, 18, 19,23, 27.

The cells (6) and (8) are in symmetric locations wrt cell (1).

Similarly, for classes #16 and #18, imagine a plane cutting through 2,5,8,11,14,17,20,23,26.

The cells (16) and (18) are in symmetric locations wrt cell (2).

So, the issue raised by witzar and me is: shouldn't we merge classes #4 and #5 because of symmetry? and shouldn't we merge classes #16 and #18 as well? These mergers would shrink your 22 classes to our 20 classes.

Yep, Captain, I hear what you guys are saying. The OP could have specified this point a bit better.

I'm a bit amazed that it made so little difference, 20 vs 22. Cool!