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Different shapes made from particular number of squares
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Given N blocks, find the number of unique shapes in a NxN blockDefinite shape of polyominosDistinct permutations of x items from n itemsLatin Squares of different sizesA starting lineup consists of 2 forwards, 2 guards and 1 center. How many different starting lineups..How many different groups of $4$ can be made from $142$?Number of ways of coloring projected faces of any hypercubeGiven N blocks, find the number of unique shapes in a NxN blockHow many different shapes that consist of five bordering squares can there be in a $3 times 3$ grid?How many $10$-letter sequences can be made from five different vowels and five different consonants?How many different words can be made?Squares sharing a side have different colors
$begingroup$
Good day! I’m currently investigating how different shapes can be made from a particular number of squares.
I have two major concerns: (1) Will there be a formula predicting the number of shapes that can be made from a certain number of squares without the flipping and rotation of formed shapes; and (2) how about with flipping and rotation?
Researching and manually listing the possible shapes that can be formed, my observations are the following:
Without flip and rotation:
Number of squares -Number of shapes made
1- 1
2- 1
3- 2
4- 5
5- 12
6- 35
7- 108
8- 384 edit: should be 369
With flip and rotation:
Number of squares- Number of shapes made
1- 1
2- 2
3- 6
4- 19
5- 63
6- 208 edit: should be 216
Any help you might extend will be highly appreciated.
PS. Will really, really be grateful if you can provide an explanation why there is (or there is no) formula that can be derived from this.
Many thanks!
Edit: Added a picture because I don't think I was able to point my message clearly :D
combinatorics
asked Sep 14 '15 at 9:46
wildberry tartwildberry tart
7927
$endgroup$
add a comment |
$begingroup$
Good day! I’m currently investigating how different shapes can be made from a particular number of squares.
I have two major concerns: (1) Will there be a formula predicting the number of shapes that can be made from a certain number of squares without the flipping and rotation of formed shapes; and (2) how about with flipping and rotation?
Researching and manually listing the possible shapes that can be formed, my observations are the following:
Without flip and rotation:
Number of squares -Number of shapes made
1- 1
2- 1
3- 2
4- 5
5- 12
6- 35
7- 108
8- 384 edit: should be 369
With flip and rotation:
Number of squares- Number of shapes made
1- 1
2- 2
3- 6
4- 19
5- 63
6- 208 edit: should be 216
Any help you might extend will be highly appreciated.
PS. Will really, really be grateful if you can provide an explanation why there is (or there is no) formula that can be derived from this.
Many thanks!
Edit: Added a picture because I don't think I was able to point my message clearly :D
combinatorics
asked Sep 14 '15 at 9:46
wildberry tartwildberry tart
7927
$endgroup$
2
$begingroup$
Look up polyominoes. I believe the 8th number in your first list should be 369, not 384.
$endgroup$
– Rory Daulton
Sep 14 '15 at 9:57
$begingroup$
What is your definition of a shape?
$endgroup$
– Aditya Agarwal
Sep 14 '15 at 9:57
$begingroup$
@AdityaAgarwal, posted the picture already. :)
$endgroup$
– wildberry tart
Sep 14 '15 at 12:28
add a comment |
$begingroup$
Good day! I’m currently investigating how different shapes can be made from a particular number of squares.
I have two major concerns: (1) Will there be a formula predicting the number of shapes that can be made from a certain number of squares without the flipping and rotation of formed shapes; and (2) how about with flipping and rotation?
Researching and manually listing the possible shapes that can be formed, my observations are the following:
Without flip and rotation:
Number of squares -Number of shapes made
1- 1
2- 1
3- 2
4- 5
5- 12
6- 35
7- 108
8- 384 edit: should be 369
With flip and rotation:
Number of squares- Number of shapes made
1- 1
2- 2
3- 6
4- 19
5- 63
6- 208 edit: should be 216
Any help you might extend will be highly appreciated.
PS. Will really, really be grateful if you can provide an explanation why there is (or there is no) formula that can be derived from this.
Many thanks!
Edit: Added a picture because I don't think I was able to point my message clearly :D
combinatorics
asked Sep 14 '15 at 9:46
wildberry tartwildberry tart
7927
$endgroup$
Good day! I’m currently investigating how different shapes can be made from a particular number of squares.
I have two major concerns: (1) Will there be a formula predicting the number of shapes that can be made from a certain number of squares without the flipping and rotation of formed shapes; and (2) how about with flipping and rotation?
Researching and manually listing the possible shapes that can be formed, my observations are the following:
Without flip and rotation:
Number of squares -Number of shapes made
1- 1
2- 1
3- 2
4- 5
5- 12
6- 35
7- 108
8- 384 edit: should be 369
With flip and rotation:
Number of squares- Number of shapes made
1- 1
2- 2
3- 6
4- 19
5- 63
6- 208 edit: should be 216
Any help you might extend will be highly appreciated.
PS. Will really, really be grateful if you can provide an explanation why there is (or there is no) formula that can be derived from this.
Many thanks!
Edit: Added a picture because I don't think I was able to point my message clearly :D
combinatorics
combinatorics
asked Sep 14 '15 at 9:46
wildberry tartwildberry tart
7927
asked Sep 14 '15 at 9:46
wildberry tartwildberry tart
7927
edited Sep 14 '15 at 12:24
wildberry tart
asked Sep 14 '15 at 9:46
wildberry tartwildberry tart
7927
asked Sep 14 '15 at 9:46
wildberry tartwildberry tart
7927
asked Sep 14 '15 at 9:46
wildberry tartwildberry tart
7927
7927
2
$begingroup$
Look up polyominoes. I believe the 8th number in your first list should be 369, not 384.
$endgroup$
– Rory Daulton
Sep 14 '15 at 9:57
$begingroup$
What is your definition of a shape?
$endgroup$
– Aditya Agarwal
Sep 14 '15 at 9:57
$begingroup$
@AdityaAgarwal, posted the picture already. :)
$endgroup$
– wildberry tart
Sep 14 '15 at 12:28
add a comment |
2
$begingroup$
Look up polyominoes. I believe the 8th number in your first list should be 369, not 384.
$endgroup$
– Rory Daulton
Sep 14 '15 at 9:57
$begingroup$
What is your definition of a shape?
$endgroup$
– Aditya Agarwal
Sep 14 '15 at 9:57
$begingroup$
@AdityaAgarwal, posted the picture already. :)
$endgroup$
– wildberry tart
Sep 14 '15 at 12:28
2
2
$begingroup$
Look up polyominoes. I believe the 8th number in your first list should be 369, not 384.
$endgroup$
– Rory Daulton
Sep 14 '15 at 9:57
$begingroup$
Look up polyominoes. I believe the 8th number in your first list should be 369, not 384.
$endgroup$
– Rory Daulton
Sep 14 '15 at 9:57
$begingroup$
What is your definition of a shape?
$endgroup$
– Aditya Agarwal
Sep 14 '15 at 9:57
$begingroup$
What is your definition of a shape?
$endgroup$
– Aditya Agarwal
Sep 14 '15 at 9:57
$begingroup$
@AdityaAgarwal, posted the picture already. :)
$endgroup$
– wildberry tart
Sep 14 '15 at 12:28
$begingroup$
@AdityaAgarwal, posted the picture already. :)
$endgroup$
– wildberry tart
Sep 14 '15 at 12:28
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
When you have such a list of integers, it's always worthwhile to search for it in the Online Encyclopedia of Integer Sequences. The two sequences you describe are OEIS sequence A000105 and OEIS sequence A001168, though it seems you got the last counts wrong and they should be $369$ and $216$, respectively.
See also MathWorld and Wikipedia on polyominoes. According to the MathWorld article, not even the growth rate is known, so no formula is known for these numbers. As to your question why this is so, that's rather hard to answer specifically. Many mathematical problems prove intractable.
answered Sep 14 '15 at 10:02
jorikijoriki
171k10190352
$endgroup$
$begingroup$
Hi! Thank you for pointing it out. I hadn't noticed that. I just manually listed the shapes so I might have been confused in the process. "no formula is known for these numbers" But might there be a pattern regarding this sequence? Or none at all? Thank you!
$endgroup$
– wildberry tart
Sep 14 '15 at 12:51
$begingroup$
@wildberrytart: I don't know of any patterns, but I wouldn't be surprised if there are some.
$endgroup$
– joriki
Sep 14 '15 at 13:05
add a comment |
$begingroup$
I thought of this question from playing blokus, and I thought I came up with an equation that works, but it only works for 1-5 pieces. I thought it was right since blokus doesn’t have pieces any larger than that. Here it is anyway though! For y= the size and x= number of pieces, $y=2^(x-1)-x+1$
answered Apr 8 at 19:19
Patrick RyanPatrick Ryan
11
New contributor
Patrick Ryan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
MathJax hint: to get multicharacter exponents, put them in braces, so 2^(x-1) gives $2^(x-1)$.
$endgroup$
– Ross Millikan
Apr 8 at 19:43
$begingroup$
Yea I didn’t know that it would actually turn it into a superscript
$endgroup$
– Patrick Ryan
Apr 8 at 21:04
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
When you have such a list of integers, it's always worthwhile to search for it in the Online Encyclopedia of Integer Sequences. The two sequences you describe are OEIS sequence A000105 and OEIS sequence A001168, though it seems you got the last counts wrong and they should be $369$ and $216$, respectively.
See also MathWorld and Wikipedia on polyominoes. According to the MathWorld article, not even the growth rate is known, so no formula is known for these numbers. As to your question why this is so, that's rather hard to answer specifically. Many mathematical problems prove intractable.
answered Sep 14 '15 at 10:02
jorikijoriki
171k10190352
$endgroup$
$begingroup$
Hi! Thank you for pointing it out. I hadn't noticed that. I just manually listed the shapes so I might have been confused in the process. "no formula is known for these numbers" But might there be a pattern regarding this sequence? Or none at all? Thank you!
$endgroup$
– wildberry tart
Sep 14 '15 at 12:51
$begingroup$
@wildberrytart: I don't know of any patterns, but I wouldn't be surprised if there are some.
$endgroup$
– joriki
Sep 14 '15 at 13:05
add a comment |
$begingroup$
When you have such a list of integers, it's always worthwhile to search for it in the Online Encyclopedia of Integer Sequences. The two sequences you describe are OEIS sequence A000105 and OEIS sequence A001168, though it seems you got the last counts wrong and they should be $369$ and $216$, respectively.
See also MathWorld and Wikipedia on polyominoes. According to the MathWorld article, not even the growth rate is known, so no formula is known for these numbers. As to your question why this is so, that's rather hard to answer specifically. Many mathematical problems prove intractable.
answered Sep 14 '15 at 10:02
jorikijoriki
171k10190352
$endgroup$
$begingroup$
Hi! Thank you for pointing it out. I hadn't noticed that. I just manually listed the shapes so I might have been confused in the process. "no formula is known for these numbers" But might there be a pattern regarding this sequence? Or none at all? Thank you!
$endgroup$
– wildberry tart
Sep 14 '15 at 12:51
$begingroup$
@wildberrytart: I don't know of any patterns, but I wouldn't be surprised if there are some.
$endgroup$
– joriki
Sep 14 '15 at 13:05
add a comment |
$begingroup$
When you have such a list of integers, it's always worthwhile to search for it in the Online Encyclopedia of Integer Sequences. The two sequences you describe are OEIS sequence A000105 and OEIS sequence A001168, though it seems you got the last counts wrong and they should be $369$ and $216$, respectively.
See also MathWorld and Wikipedia on polyominoes. According to the MathWorld article, not even the growth rate is known, so no formula is known for these numbers. As to your question why this is so, that's rather hard to answer specifically. Many mathematical problems prove intractable.
answered Sep 14 '15 at 10:02
jorikijoriki
171k10190352
$endgroup$
When you have such a list of integers, it's always worthwhile to search for it in the Online Encyclopedia of Integer Sequences. The two sequences you describe are OEIS sequence A000105 and OEIS sequence A001168, though it seems you got the last counts wrong and they should be $369$ and $216$, respectively.
See also MathWorld and Wikipedia on polyominoes. According to the MathWorld article, not even the growth rate is known, so no formula is known for these numbers. As to your question why this is so, that's rather hard to answer specifically. Many mathematical problems prove intractable.
answered Sep 14 '15 at 10:02
jorikijoriki
171k10190352
answered Sep 14 '15 at 10:02
jorikijoriki
171k10190352
answered Sep 14 '15 at 10:02
jorikijoriki
171k10190352
answered Sep 14 '15 at 10:02
jorikijoriki
171k10190352
171k10190352
$begingroup$
Hi! Thank you for pointing it out. I hadn't noticed that. I just manually listed the shapes so I might have been confused in the process. "no formula is known for these numbers" But might there be a pattern regarding this sequence? Or none at all? Thank you!
$endgroup$
– wildberry tart
Sep 14 '15 at 12:51
$begingroup$
@wildberrytart: I don't know of any patterns, but I wouldn't be surprised if there are some.
$endgroup$
– joriki
Sep 14 '15 at 13:05
add a comment |
$begingroup$
Hi! Thank you for pointing it out. I hadn't noticed that. I just manually listed the shapes so I might have been confused in the process. "no formula is known for these numbers" But might there be a pattern regarding this sequence? Or none at all? Thank you!
$endgroup$
– wildberry tart
Sep 14 '15 at 12:51
$begingroup$
@wildberrytart: I don't know of any patterns, but I wouldn't be surprised if there are some.
$endgroup$
– joriki
Sep 14 '15 at 13:05
$begingroup$
Hi! Thank you for pointing it out. I hadn't noticed that. I just manually listed the shapes so I might have been confused in the process. "no formula is known for these numbers" But might there be a pattern regarding this sequence? Or none at all? Thank you!
$endgroup$
– wildberry tart
Sep 14 '15 at 12:51
$begingroup$
Hi! Thank you for pointing it out. I hadn't noticed that. I just manually listed the shapes so I might have been confused in the process. "no formula is known for these numbers" But might there be a pattern regarding this sequence? Or none at all? Thank you!
$endgroup$
– wildberry tart
Sep 14 '15 at 12:51
$begingroup$
@wildberrytart: I don't know of any patterns, but I wouldn't be surprised if there are some.
$endgroup$
– joriki
Sep 14 '15 at 13:05
$begingroup$
@wildberrytart: I don't know of any patterns, but I wouldn't be surprised if there are some.
$endgroup$
– joriki
Sep 14 '15 at 13:05
add a comment |
$begingroup$
I thought of this question from playing blokus, and I thought I came up with an equation that works, but it only works for 1-5 pieces. I thought it was right since blokus doesn’t have pieces any larger than that. Here it is anyway though! For y= the size and x= number of pieces, $y=2^(x-1)-x+1$
answered Apr 8 at 19:19
Patrick RyanPatrick Ryan
11
New contributor
Patrick Ryan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
MathJax hint: to get multicharacter exponents, put them in braces, so 2^(x-1) gives $2^(x-1)$.
$endgroup$
– Ross Millikan
Apr 8 at 19:43
$begingroup$
Yea I didn’t know that it would actually turn it into a superscript
$endgroup$
– Patrick Ryan
Apr 8 at 21:04
add a comment |
$begingroup$
I thought of this question from playing blokus, and I thought I came up with an equation that works, but it only works for 1-5 pieces. I thought it was right since blokus doesn’t have pieces any larger than that. Here it is anyway though! For y= the size and x= number of pieces, $y=2^(x-1)-x+1$
answered Apr 8 at 19:19
Patrick RyanPatrick Ryan
11
New contributor
Patrick Ryan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
MathJax hint: to get multicharacter exponents, put them in braces, so 2^(x-1) gives $2^(x-1)$.
$endgroup$
– Ross Millikan
Apr 8 at 19:43
$begingroup$
Yea I didn’t know that it would actually turn it into a superscript
$endgroup$
– Patrick Ryan
Apr 8 at 21:04
add a comment |
$begingroup$
I thought of this question from playing blokus, and I thought I came up with an equation that works, but it only works for 1-5 pieces. I thought it was right since blokus doesn’t have pieces any larger than that. Here it is anyway though! For y= the size and x= number of pieces, $y=2^(x-1)-x+1$
answered Apr 8 at 19:19
Patrick RyanPatrick Ryan
11
New contributor
Patrick Ryan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I thought of this question from playing blokus, and I thought I came up with an equation that works, but it only works for 1-5 pieces. I thought it was right since blokus doesn’t have pieces any larger than that. Here it is anyway though! For y= the size and x= number of pieces, $y=2^(x-1)-x+1$
answered Apr 8 at 19:19
Patrick RyanPatrick Ryan
11
New contributor
Patrick Ryan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Apr 8 at 21:05
answered Apr 8 at 19:19
Patrick RyanPatrick Ryan
11
New contributor
Patrick Ryan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Apr 8 at 19:19
Patrick RyanPatrick Ryan
11
answered Apr 8 at 19:19
Patrick RyanPatrick Ryan
11
11
New contributor
Patrick Ryan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Patrick Ryan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Patrick Ryan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
MathJax hint: to get multicharacter exponents, put them in braces, so 2^(x-1) gives $2^(x-1)$.
$endgroup$
– Ross Millikan
Apr 8 at 19:43
$begingroup$
Yea I didn’t know that it would actually turn it into a superscript
$endgroup$
– Patrick Ryan
Apr 8 at 21:04
add a comment |
$begingroup$
MathJax hint: to get multicharacter exponents, put them in braces, so 2^(x-1) gives $2^(x-1)$.
$endgroup$
– Ross Millikan
Apr 8 at 19:43
$begingroup$
Yea I didn’t know that it would actually turn it into a superscript
$endgroup$
– Patrick Ryan
Apr 8 at 21:04
$begingroup$
MathJax hint: to get multicharacter exponents, put them in braces, so 2^(x-1) gives $2^(x-1)$.
$endgroup$
– Ross Millikan
Apr 8 at 19:43
$begingroup$
MathJax hint: to get multicharacter exponents, put them in braces, so 2^(x-1) gives $2^(x-1)$.
$endgroup$
– Ross Millikan
Apr 8 at 19:43
$begingroup$
Yea I didn’t know that it would actually turn it into a superscript
$endgroup$
– Patrick Ryan
Apr 8 at 21:04
$begingroup$
Yea I didn’t know that it would actually turn it into a superscript
$endgroup$
– Patrick Ryan
Apr 8 at 21:04
add a comment |
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2
$begingroup$
Look up polyominoes. I believe the 8th number in your first list should be 369, not 384.
$endgroup$
– Rory Daulton
Sep 14 '15 at 9:57
$begingroup$
What is your definition of a shape?
$endgroup$
– Aditya Agarwal
Sep 14 '15 at 9:57
$begingroup$
@AdityaAgarwal, posted the picture already. :)
$endgroup$
– wildberry tart
Sep 14 '15 at 12:28