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Topic: Hexomino


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In the News (Fri 1 Aug 14)

  
  Hexomino - Wikipedia, the free encyclopedia
As with other polyominoes, rotations and reflections of a hexomino are not considered to be distinct shapes and with this convention, there are thirty-five different hexominoes.
2 hexominoes (coloured green) have an axis of mirror symmetry at 45° to the gridlines.
If the hexominoes are placed on a checkerboard pattern, then 11 of the hexominoes will cover an even number of fl squares (either 2 white and 4 fl or vice-versa) and 24 of the hexominoes will cover an odd number of fl squares (3 white and 3 fl).
en.wikipedia.org /wiki/Hexomino   (476 words)

  
 Hexomino Constructions
This requires a sixfold replica of the hexomino to be built with the sextuplicated piece used twice in the solution.
Here the problem is to create an hexomino construction with a hole of a similar shape.
The 3x120 is not possible as can be seen from the diagram below where one of the hexominoes must divide the rectangle into two parts neither of which contains a multiple of six squares.
www.geocities.com /alclarke0/PolyPages/hexopatts.htm   (793 words)

  
 Primes of the D hexomino   (Site not responding. Last check: 2007-10-18)
Any rectangle tiled by the D hexomino has one side divisible by 6.
However it is placed, a D hexomino covers a total of 0.
It is easy to check that a (6m + 2) × (6n + 3) rectangle can be placed so that it covers a non-zero total, and the same for a (6m + 4) × (6n + 3) rectangle.
www.math.ucf.edu /~reid/Polyomino/d6_rect.html   (172 words)

  
 Primes of the Y hexomino   (Site not responding. Last check: 2007-10-18)
If the Y hexomino tiles a rectangle with an odd side, then the other side is divisible by 8.
No matter how it is placed, each Y hexomino covers a total of 2 or -2.
However, it would be tiled by an even number of Y hexominoes, which would cover a total that is divisible by 4, a contradiction.
www.math.ucf.edu /~reid/Polyomino/y6_rect.html   (190 words)

  
 Hexomino -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-18)
The figure shows all possible hexominoes, and are coloured according to their (Click link for more info and facts about symmetry group) symmetry groups:
With checkerboard colouring, it has 106 white and 104 fl squares (or vice versa), so parity does not prevent a packing, and a packing is indeed possible -- see.
is a page by Jürgen Köller on hexominos, including symmetry, packing and other aspects.
www.absoluteastronomy.com /encyclopedia/H/He/Hexomino.htm   (304 words)

  
 Hexominos
Hexominos are figures you can form by six squares.
If you colour a figure alternately fl and white there are 24 "odd" hexominos with three fl and three white squares and 11 "even" hexominos with four fl and two white squares.
As is in the case of pentominos there are innumerable new figures formed of hexominos.
www.mathematische-basteleien.de /hexominos.htm   (405 words)

  
 Hexominoes   (Site not responding. Last check: 2007-10-18)
Hexomino From MathWorld A hexomino is a 6-polyomino.
Hexominoes The 18x14 rectangle below was constructed from the 35 hexominoes by Michael Keller.
Hexominoes The groups use grid paper and color tiles to create as many different hexominoes as they can.
www.education-resource-finds.com /directory/geometry/hexominoes.html   (473 words)

  
 Part II   (Site not responding. Last check: 2007-10-18)
By statement 1, we need not consider those of the hexominoes, so all the rest of the tetrapleura are projective duals of pentominoes.
The complete projective duals of the first seven hexominoes on the bottom row have one enclosed, finite region and six edges each (since each of those hexominoes has 1 2x2 square and no holes), and so any incomplete projective duals formed from them must have one edge deleted.
This raises the number of pentapleura to 49 (28+4+4+4+3+2+2+2), and we still have to consider the incomplete projective duals of the O hexomino.
www.saintanns.k12.ny.us /depart/math/Lawrence/part2.html   (869 words)

  
 Teaching Math: Grades 3-5: Connections
I remembered when we made hexominos and tried to make all the ones with four in a row again.
I found seven hexominos with a line of 4 squares.
I tried to remember some of the hexominos that we made before, and thought about how we had folded them to see if they made cubes.
www.learner.org /channel/courses/teachingmath/grades3_5/session_06/section_01_d.html   (572 words)

  
 Mathpuzzle.com
The cracker was circular and had 21 holes arranged in a square grid minus the corners.
I wondered if there was a hexomino with unit squares the size of the grid, three copies of which could cover 18 holes orthogonally.
I found one that happens to be a path hexomino.
www.mathpuzzle.com /21Nov04.html   (3362 words)

  
 Programs - Games & Tests
The program is a game in which the user must fit all the pieces into a rectangle.
Tetramino, pentomino, hexomino and heptomino puzzles are included.
I haven't found any solution for hexominoes yet (there must be tones of such stuff on the Internet, but I'm too lazy to look for it).
www.angelfire.com /home/florinleon/prog_gt.htm   (458 words)

  
 One-sided hexomino similar hole problems
The possible constructions are based on factorisations of 360 (the total area of the pieces) in the form (a² - b²) x cd.
Four quadruplicated hexominoes - 35 problems (61 if we consider solutions in mirror pairs as in some solutions below).
The two at the bottom right are not possible owing to their being unbalanced.
clarkjag.idx.com.au /PolyPages/1s6oSH.htm   (273 words)

  
 Teaching Math: Grades 3-5: Connections
While sorting hexominos, a group of fifth-graders noticed that three of their hexominos could be folded to make a cube and the others could not.
Whitney: Well, if it was like our Z-shaped hexomino, they might be harder to cut out and to keep neatly on a shelf before you fold them.
But most of the hexominos have faces that overlap when you try to make a box, so they don't make a six-sided cube.
www.learner.org /channel/courses/teachingmath/grades3_5/session_06/section_01_f.html   (578 words)

  
 LIFELINE: Number 1/March 1971   (Site not responding. Last check: 2007-10-18)
At that time he had followed the life histories of all but one of the pentominoes, all but one of the hexominoes, and all but seven of the heptominoes.
By now we all know the fate of the notorious R-pentomino which, in its first generation, becomes a hexomino (the one who's fate was unknown to Conway).
This leaves us with the seven 'unknown' heptominoes shown here which Conway arbitrarily labeled B, C, D, E, F, H, and I. Heptomino B whose first generation appears in the 29th generation of the R-pentomino eventually becomes three blocks, one ship, and two gliders after 148 generations - so its history is known.
members.aol.com /life1ine/life/page1.htm   (493 words)

  
 Isoperimetric Polyominoes Solutions
Create an area of 10 square units, produce a copy, a duplication and a triplication.
With the whole set produce simultaneous triplication and quadruplication of an hexomino.
Make a 12x13 rectangle with a hole in the shape of an hexomino.
clarkjag.idx.com.au /PolyPages/Isopolyosols.html   (269 words)

  
 Math Magic
George Sicherman first solved the T pentomino, the S hexomino, and the vast majority of the heptominoes and octominoes!
His results are here for tetrominoes, pentominoes, and hexominoes.
The yellow figures are the original problem, the orange is Corey's variant, the blue are Richard's variant, and the purple is both sets of rules.
www.stetson.edu /~efriedma/mathmagic/1104.html   (726 words)

  
 Puzzle Corner
This was in a team match, and at the other table they bid 7NT and went down, so we cleared a bundle of IMPs!' So much for his worst luck.' Can you construct such a deal?"
The five tiles are identical in size and shape but may be turned over so that some are mirror images of the others.
Walter Cluet asks, the fifth letter of the alphabet is to the sixth and the 18th is to the 16th as the 12th is to?
www.techreview.com /articles/04/03/puzzle0304.asp?p=0   (1626 words)

  
 Untitled Document   (Site not responding. Last check: 2007-10-18)
There are three hexominos which can do it, all path hexominos, one in two ways.
I was able to find several different hexominoes that could be placed in the 5x5 minus corners grid:
On the right is the hexomino used in the solution I found more recently.
www.mathpuzzle.com /CrackedCracker.htm   (122 words)

  
 Hexominos   (Site not responding. Last check: 2007-10-18)
What is a hexomino and how many different shapes are possible?
This reference has a list of the 12 possible pentomino but It doesn't have the 35 hexominos.
I did find a diagram containing the 35 hexominos at http://members.aol.com/wgreview/hexomino.html
mathcentral.uregina.ca /QQ/database/QQ.09.00/tom2.html   (59 words)

  
 Blocking Polyominoes   (Site not responding. Last check: 2007-10-18)
The only condition is that if we take out 1 piece there could not be part of the figure still blocked.
The hexominoes of N=4 would not be solution because we can move together the 2 blue pieces to the left:
So to be a good solution to this puzzle we cannot move 1 or more pieces of the figure.
www.eldar.org /~problemi/pfun/blocked.html   (288 words)

  
 Puzzles -- LEGO
A 'pentomino' is a shape created by adjoining 5 (penta-) squares together edge to edge.
A 'polyomino' is a generalization which incorporates any number of squares (not just 5, for example, a 'hexomino' is a shape built from 6 squares).
Finally, a 'polycube' is a similar shape, but one which is not restricted to two dimensions...
www.ericharshbarger.org /lego/puzzles.html   (1620 words)

  
 The Geometry Junkyard: Unfolded Polyhedra
A common way of making models of polyhedra is to unfold the faces into a planar pattern, cut the pattern out of paper, and fold it back up.
This Geometry Forum problem of the week asks for the number of different hexominoes, and for how many of them can be folded into a cube.
It turns out that the familiar cross hexomino pattern for folding cubes can also be used to fold three other polyhedra with four, five, and eight sides.
www.ics.uci.edu /~eppstein/junkyard/unfold.html   (717 words)

  
 Pentomino Constructions   (Site not responding. Last check: 2007-10-18)
Maximum number of holes in a hexomino construction
Multiple replications of a one-sided hexomino based on 2-2-2-2-2-2-2-4-4 or 2-2-2-2-2-2-3-3-3-3.
Other constructions with the 363 octominoes without a hole.
clarkjag.idx.com.au /PolyPages/Problems.htm   (121 words)

  
 Cornucopia   (Site not responding. Last check: 2007-10-18)
If you take the thirty-five possible hexominoes and eliminate those wth symmetry and those which are non-elongate (containing a 2X2 square), seventeen remain.
Using Bill Cutler's box-filling program, Walter found eight solutions where the extra 2X2 square filled other than the center location and twenty-nine solutions where the 1X1 squares are in locations other than the corners.
The ten hexomino pieces also make a 6X10 rectangle.
www.johnrausch.com /Puzzleworld/puz/cornucopia.htm   (171 words)

  
 Inleiding   (Site not responding. Last check: 2007-10-18)
Each type of polyomino is named according to how many squares are used to make it.
So there are monominoes (1 square only), dominoes (2 squares), triominoes (3 squares), tetrominoes (4 squares), pentominoes (5 squares), hexominoes (6 squares) and so on.
Though the idea of such shapes has been around in Recreational Mathematics since the beginning of the 1900's, it was not until the latter half of the century that they became as popular as they are today.
www.ping.be /~demeod/inleidinge.html   (276 words)

  
 Math Forum: Math 7 - Alejandre
He also loaned me a book titled Polyominoes by Solomon W. Golomb.
To make more connections to a variety of mathematics I decided to write a domino activity to replace the hexomino activity presented in the text.
The hexomino activity might be appropriate as a follow-up homework assignment.
www.mathforum.com /alejandre/frisbie/poly.html   (501 words)

  
 Math Lair - Tic-tac-toe   (Site not responding. Last check: 2007-10-18)
As a matter of fact, most polyominoes of size 5 or greater are losers.
There are only three pentominoes and (probably) one hexomino that are winners:
Since almost every heptomino (size 7) or larger will contain at least two different hexominoes, and there is only one winning hexomino, it is not too hard to show that all 107 order-7 animals contain a smaller loser and thus are losers themselves.
www.stormloader.com /ajy/tictactoe.html   (841 words)

  
 The Life of Games, page 32 - Narrow Passage, a hexomino puzzle and contest   (Site not responding. Last check: 2007-10-18)
The 35 hexominoes shown above (you can use a Sextillions set without the duplicate piece) can form a fully enclosed long passage no wider than one square at any point.
The passage can twist and turn in any direction and even loop back on itself without intersecting.
Notice how the surrounding pieces must be connected on at least one full edge, not merely corners.
www.gamepuzzles.com /tlog/tlog32.htm   (197 words)

  
 Citations: Tiling rectangles with polyominoes - Golomb (ResearchIndex)   (Site not responding. Last check: 2007-10-18)
The boot tetromino (T tetromino) is well known to have rectangular order 4.
Klarner [12] showed that the boot hexomino has rectangular order 18, and Golomb [8] showed that the....
Klarner [12] showed that the boot hexomino has rectangular order 18, and Golomb [8] showed that the boot polyomino of size 8n 4 has rectangular order 4n 4.
citeseer.ist.psu.edu /context/485360/0   (465 words)

  
 Polyominoes
Unfortunately, as with the tetrominoes, an odd number of these have a 4-2 chessboard colouring and so no rectangle can be filled with all 35.
The 11 hexominoes with a 4-2 colouring are shown in the right hand part of the figure below where the remaining 24 pieces form two 8x9 rectangles.
The one-sided trominoes, tetrominoes, pentominoes and hexominoes cover a total area equal to that of a side 22 square.
clarkjag.idx.com.au /PolyPages/Polyominoes.html   (1022 words)

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