[math-fun] Packing puzzle
Consider a 3-D tile made of 5 cubies arranged like a P-pentomino, 1 cubie thick. You pack these into a 3x3 cube (together they fill all but 2 cubies). Suppose one of the unpacked cubies is the center cubie. Can the remaining unpacked cubie be at a vertex of the 3x3? An edge? A face center?
Right... What's a cubie? On Wed, Aug 28, 2013 at 6:19 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a 3-D tile made of 5 cubies arranged like a P-pentomino, 1 cubie thick.
You pack these into a 3x3 cube (together they fill all but 2 cubies).
Suppose one of the unpacked cubies is the center cubie.
Can the remaining unpacked cubie be at a vertex of the 3x3? An edge? A face center?
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
I was borrowing from Rubik's cube terminology. A cubie is one of the 27 subcubes of the 3x3 cube. I'm sure there's more standard terminology, I just refuse to use it.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Mike Stay Sent: Wednesday, August 28, 2013 9:30 PM To: math-fun Subject: Re: [math-fun] Packing puzzle
Right... What's a cubie?
On Wed, Aug 28, 2013 at 6:19 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a 3-D tile made of 5 cubies arranged like a P-pentomino, 1 cubie thick.
You pack these into a 3x3 cube (together they fill all but 2 cubies).
Suppose one of the unpacked cubies is the center cubie.
Can the remaining unpacked cubie be at a vertex of the 3x3? An edge? A face center?
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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a Rubik cube contains 27 cubies (one of which is hard to see) On Wed, Aug 28, 2013 at 9:30 PM, Mike Stay <metaweta@gmail.com> wrote:
Right... What's a cubie?
On Wed, Aug 28, 2013 at 6:19 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a 3-D tile made of 5 cubies arranged like a P-pentomino, 1 cubie thick.
You pack these into a 3x3 cube (together they fill all but 2 cubies).
Suppose one of the unpacked cubies is the center cubie.
Can the remaining unpacked cubie be at a vertex of the 3x3? An edge? A face center?
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
Everybody missed Mike's Bill Cosby homage... On Wed, Aug 28, 2013 at 9:40 PM, Neil Sloane <njasloane@gmail.com> wrote:
a Rubik cube contains 27 cubies (one of which is hard to see)
On Wed, Aug 28, 2013 at 9:30 PM, Mike Stay <metaweta@gmail.com> wrote:
Right... What's a cubie?
On Wed, Aug 28, 2013 at 6:19 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a 3-D tile made of 5 cubies arranged like a P-pentomino, 1 cubie thick.
You pack these into a 3x3 cube (together they fill all but 2 cubies).
Suppose one of the unpacked cubies is the center cubie.
Can the remaining unpacked cubie be at a vertex of the 3x3? An edge? A face center?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
http://en.wikipedia.org/wiki/Cubit http://en.wiktionary.org/wiki/cubie On Wed, Aug 28, 2013 at 9:11 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Everybody missed Mike's Bill Cosby homage...
On Wed, Aug 28, 2013 at 9:40 PM, Neil Sloane <njasloane@gmail.com> wrote:
a Rubik cube contains 27 cubies (one of which is hard to see)
On Wed, Aug 28, 2013 at 9:30 PM, Mike Stay <metaweta@gmail.com> wrote:
Right... What's a cubie?
On Wed, Aug 28, 2013 at 6:19 PM, David Wilson < davidwwilson@comcast.net> wrote:
Consider a 3-D tile made of 5 cubies arranged like a P-pentomino, 1 cubie thick.
You pack these into a 3x3 cube (together they fill all but 2 cubies).
Suppose one of the unpacked cubies is the center cubie.
Can the remaining unpacked cubie be at a vertex of the 3x3? An edge? A face center?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Not everyone. You got it. But seriously, God didn't even remember what a cubie was.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Wednesday, August 28, 2013 10:12 PM To: math-fun Subject: Re: [math-fun] Packing puzzle
Everybody missed Mike's Bill Cosby homage...
On Wed, Aug 28, 2013 at 9:40 PM, Neil Sloane <njasloane@gmail.com> wrote:
a Rubik cube contains 27 cubies (one of which is hard to see)
On Wed, Aug 28, 2013 at 9:30 PM, Mike Stay <metaweta@gmail.com> wrote:
Right... What's a cubie?
On Wed, Aug 28, 2013 at 6:19 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a 3-D tile made of 5 cubies arranged like a P-pentomino, 1 cubie thick.
You pack these into a 3x3 cube (together they fill all but 2 cubies).
Suppose one of the unpacked cubies is the center cubie.
Can the remaining unpacked cubie be at a vertex of the 3x3? An edge? A face center?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 8/28/2013 8:19 PM, David Wilson wrote:
Consider a 3-D tile made of 5 cubies arranged like a P-pentomino, 1 cubie thick.
You pack these into a 3x3 cube (together they fill all but 2 cubies).
Suppose one of the unpacked cubies is the center cubie.
Can the remaining unpacked cubie be at a vertex of the 3x3? An edge? A face center?
A pleasant little puzzle to solve at the end of a long day. (BTW, anyone wondering what Allan meant can find the Bill Cosby routine here: http://www.youtube.com/watch?v=bputeFGXEjA ) Spoiler below: A P-pentomino is planar, so each tile lies within a one-cubie-thick slice of the cube. Since the center is unused, such a slice consists of the nine cubies on one face of the cube. The face center cubies each belong to only one face, so the five tiles can only use five of them. Thus one face center must remain unused. It remains to show that such a packing is possible. Here's one, layer-by-layer: 2 2 1 3 1 1 3 1 1 2 2 2 3 x 4 3 x 4 5 5 4 5 5 4 3 5 4 -- Fred W. Helenius fredh@ix.netcom.com
Glad you liked it. If you believe it, I came up with this while lying in bed staring into the darkness. I was even able to tile the cube in my head, so I knew there was a solution. I don't think I could solve the Soma cube that way.
participants (6)
-
Allan Wechsler -
David Wilson -
Fred W. Helenius -
James Buddenhagen -
Mike Stay -
Neil Sloane