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with John Gough < [email protected]> Gemblo and Polyhex Blokus Blokus joined together whole-edge to whole-edge. Gemblo is one of the possible polyhex Blokus is an abstract strategy board game games that might be invented. Why for 2 to 4 players. It uses polyominoes obvious? Because both Blokus and Blokus WHicH are plane geometric lgures formed Trigon used families of shapes made from by joining one or more equal squares edge regular polygons that tessellate. How many to edge refer &igure . 0olyominoes HaVe are there? been described as being “a polyform whose cells are squaresv and they are classiled Gemblo according to how many cells they have. ie. number of cells. Wikipedia, 2017 March In 2005, Gemblo was created by Justin Oh 19). Polyominoes and other mathematically in Korea. It is an abstract strategy board patterned families of shapes were invented game with translucent, coloured pieces, by Solomon Golomb, and popularised by each of which is made up of one to lve hex- Martin Gardner. agons. The six colours are clear, red, yellow, green, blue, and purple. Despite the obvious similarities to Blokus, Oh has stated that when he was inventing Gemblo he did not know about Blokus. The name is based on the word ‘gem’, alluding to the gem-like colours used for the play- ing pieces and ‘blocking’—a central feature of playing. The size of the playing board depends on the number of people playing. Figure 1: Examples of polyominoes With players using a hexagonal shaped board that has sides of 8 unit-hexagons. Blokus—using squares—has a sister- game called Blokus Trigon that uses poly- Playing Gemblo iamonds, those families of shapes made s Gemblo can be played as a one- from unit-sided equilateral triangles joined person puzzle or solitaire, or by whole-edge to whole-edge. Polyiamonds multiple players ie. 2, , 4, or were named and popularised by Thomas H. persons). A complicated version O’Beirne in articles on recreational mathe- for 5 players also exists. matics in New Scientist 191. s In the two-player version of the game, As soon as Blokus appeared it became players use the rules for a stand- apparent that there ought to be a strategy ard 4-player game, with each player board game based on polyhexes, those taking a set of polyhex pieces of families of shapes made from unit-hexagons two colours, and alternating turns: 32 amt 73(3) 2017 Player-1-colour-A, Player-2-colour-B, .o stackingÐ) The lrst piece must Player-1-colour-C, and so on. have one of its unit-hexagons s With , 4 or players, each player covering the unit-hexagon speciled uses a set of identical polyhexes in as the start for that player. one unique colour. s In the six-player version, each player’s s Alternatively, three players could starting hexagon is one of the corners follow the 2-player approach, with of the large board. each player taking a set of polyhex s %ach piece that a player places after pieces of two colours and in a play- his or her lrst piece cannot touch, er’s successive turns placing a piece either edge to edge, or vertex to ver- of one, and then the other, of their tex, any previously-placed pieces of two colours. the same colour. Instead each succes- s %xcept in the case of the two-player sive piece placed by a player must be version, each player has an identical a distance of 1 from another piece of set of eighteen polyhex pieces in the same colour, where 1 is the length a unique colour: 8 pentahexes; 5 of a side of one of the small unit-hex- tetrahexes; 3 trihexes; 1 duohex; agons. That is, there is always a gap and, 1 monohex. of a unit-length between pieces of the s The goal is to get rid of all, or end same colour. We can think of that ‘1’ with as few as possible, of your as a unit-length bridge that implicitly pieces by placing them on the joins two pieces of the same colour: hexagonal game board. the rules allow two pieces of the s Players take turns. In each turn, same colour to be implicitly joined a player chooses one of his or her by several unit-length bridges. Pieces unused pieces, and places it on of different colours may abut against the board covering empty spaces. each other. Figure 2: Example of the 18 Gemblo polyhex shapes in one colour. amt 73(3) 2017 33 The unit-length joining distance Problems to ponder from playing between two pieces of the same colour Gemblo cannot directly cross through another Some possible problems to consider during player’s piece, but it can pass between and/or after playing the Gemblo game with the edge-against-edge junction of two either a single or multiple players could pieces of one or more other players. include: 1. What is the total number of unit- hexagons used to make the six coloured sets of 18 pieces? B 2. a) Is 18, the number of Gemblo A A pieces of one colour, the total A A B B number of geometrically distinct polyhexes that can be made using B A B up to lve unit-hexes? b) If not, how many distinct poly- A hexes can be made using up to lve A unit-hexes, joined whole-edge to whole-edge? 3. What is the area of the 8-hexagon- Figure 3: Not a legal move—the ‘join’ passes through sided board, measured in unit- red (Scooter Games, 2013, January 31). hexagons? 4. a) Can all the playing pieces be ltted together, with no gaps or holes, in a hexagonal styled board? B A A b) If so, what is the size of that A A B board? B 5. What mathematically distinct B shapes can be made joining together B tessellating) the 18 polyhexes of one A colour without holes? A . a) What mathematically distinct shapes can be made joining together A the 18 polyhexes of one colour with unit-hexagon holes contained within Figure 4: A legal move—observe the join (Scooter the whole conlguration? Games, 2013, January 31). 7. b) What overall combined shape The game ends when all players have contains a maximum number placed as many pieces as they can. Players of unit-hexagon holes? then count the number of hexagonal units 8. Using the 18 polyhexes of one in the pieces they could not place, and the colour: player with the lowest score wins. a) What is the maximum area There are further rules for deciding who measured in unit-hexagons) wins in the case of a draw between lowest that can be continuously bound- scores, and specifying the starting position ed—’fenced’—by the 18 pieces, for each player’s lrst piece. joining them together, whole-edge to whole-edge? b) What is the minimum area that can be fenced? 34 amt 73(3) 2017 Blokus variants Wikipedia. 2017, July 22). Blokus. Retrieved 25 Use the Gemblo pieces and board to play July 2017 from https://en.wikipedia.org/wiki/ Blokus with the Blokus rules Wikipedia, 2017 Wikipedia. 2017, March 19). Polyomino. Retrieved July 22). 25 July 2017 from https://en.wikipedia.org/ s After the lrst piece has been played, wiki/Polyomino the following turns by players will World of Board Games ,,P. n.d.). Gemblo. require them to place pieces so that Retrieved 25 July 2017 from http://www.world- of-board-games.com.sg/docs/Gemblo.pdf they do not touch another piece of the same colour, but only at one or more vertices. Further reading s Pieces of the same colour must not Koster, R. 200, September 2). Raph Koster’s Website: Gemblo-if-you-like-blokus. Retrieved have adjacent edges. 25 July 2017 from http://www.raphkoster. com/200/09/2/Gemblo-if-you-like-blokus/ References Wikipedia. 2017, June 24). Multiple Discovery. BoardGameGeek. n.d.). Gemblo. Retrieved 25 Retrieved 25 July 2017 from https:// July 2017 from http://boardgamegeek.com/ en.wikipedia.org/wiki/Multiple?discovery boardgame/19427/Gemblo The Poly Pages. n.d.). Polyhexes. Retrieved 25 Scooter Games. 2013, January 31). Gemblo. July 2017 from http://www.recmath.com/ Retrieved 25 July 2017 from http:// PolyPages/PolyPages/index.htm?Polyhexes.html scootergames.wordpress.com/2013/01/31/ Gemblo/ amt 73(3) 2017 35.
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