Jg Space Games
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John Gough — Spatial Thinking Dice Games — 2004
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Playing With Space: Dice Games, and Others John Gough — [email protected] Introduction Most of the following games require pencil and paper, several ordinary sixsided dice of different colors, dotty or square grid paper, or a geoboard. Each game is designed to provide gamebased experiences that can help develop ideas about space, geometry, and concepts such as vertex, side, face, perimeter, area, surface area, volume, polygon, and polyhedra. Most of the games can be played with just two players. It is also useful to play together in pairs, with two or more pairs of players competing. Some games can be easily adapted for play by a whole class, with each person either playing individually, or as one of a pair. Make a Cube Players aim to score by building successive complete layers of a 3x3x3 cubic unit cube. Equipment A 3x3 square grid baseboard (the size of the squares fits the size of the cubes used in playing: for example, centicubes need a 3cm x3cm grid). An ordinary dice. Centicubes or other fittogether unit cubes, or Cuisenaire, or MAB materials. Pencil and paper for scoring. Number of Players 2 players Playing Players take turns. In each turn, a player rolls the dice. Depending on the result the player takes the appropriate piece or pieces: 1 = unit cube 4 = two unit cubes 2 = two cubes together, in a “stick” 5 = two lots of two cubes in a “stick” 3 = three cubes in a “stick” 6 = two lots of three cubes in a “stick” Then the player places the piece or pieces anywhere on the playing board, so that: — each piece fits in the 3x3 base (no overhanging) — this does not leave any unfilled holes in the 3x3x3 cube that is being built. Scoring A player scores 1 point for completing a whole layer (that is, placing one or more pieces that completes either the first or second “floor” of the 3x3x3 cube that is being built; this
John Gough — Spatial Thinking Dice Games — 2004
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may mean that some part of the piece or pieces used to complete the “floor” will protrude vertically above the “floor”.) The player who completes the third “floor” (that is, finishes the whole 3x3x3 cube) scores 5 points, minus each extra cube that protrudes above the level of the third “floor”. Play continues, through successive rounds of starting at “ground level”, and building the 3x3x3 cube, until one player reaches a total score of 20 points. Or the player with the highest score after an agreed time wins the game.
John Gough — Spatial Thinking Dice Games — 2004
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Polygon Factory Players use dice rolls and playing to get a maximum score as they make collections of polygons. Equipment Pencil and paper. Four ordinary dice: Red, Blue, Yellow and Green. A Player’s Table for each player. Number of Players 2 or more players Rules for Play Players aim to use the results of dice rolling, and other actions, to make a particular set of polygons, using “material” that is specified by the results of the dice rolling (which will be explained). Players take turns. In each turn a player may do ONE of the following: • roll some or all of the dice; • request to “buy” some of the “material” belonging to another player; • request to “swap”, in some negotiated way, some of his or her “material” for some of another player’s “material”; or • declare that one of a set of polygons, or a complete set of polygons, has been completed. Red dice: the result (1 to 6) specifies the QUANTITY of ONE of the other dice. Blue dice: the result (1 to 6) specifies the LENGTH of a side for a possible polygon. Yellow dice: the result (1 to 6) specifies the ANGLEsize: 1 = 20o, 2 = 30 o, 3 = 45 o, 4 = 60 o, 5 = 90 o, and 6 = 108 o. Green dice: the result (1 to 6) specifies the POLYGONtype (see the Player’s Table, below). Rolling When a player rolls the dice, the player chooses which of ONE the other dice will be selected to be combined with the Red Quantity. For example, a player might roll and obtain the following result: Red = 4, Blue = 2, Yellow = 5, and Green = 2. Hence the player can specify, that is, tally, in his or her Player’s Table: — four sides of length 2 units; or
John Gough — Spatial Thinking Dice Games — 2004
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— four angles of 90 degrees; or — four polygons which are “digons”, that is, twosided figures, otherwise known as “angles”. If a player rolls the dice and gets 1 for Green (Type of Polygon), this result is “wild”, and the player can freely use this as though it were any other dice result (that is, the player can freely use this as any other type of polygon). “Buying” If a player “buys” tallied material from another player, the other player receives a payment or bonus score of 1, and the buyer loses 1 point of score. “Swapping” If two players agree to “swap” some their materials, then each scores a bonus of 1 point. Player’s Table Possible Polygon Sets: — one of each kind: e.g. a family of triangles consists of all acuteangle scalene (three different lengths, and three different acute angles), one obtuseangle scalene, isosceles nonright, rightisosceles, equilateral. — a family of one kind: e.g. a family of angles consists of acute, right, obtuse, and reflex. BLUE Length of Side | 1 | 2 | 3 | 4 | 5 | 6 | Quantity of length | | | | | | | YELLOW o o o o o Size of Angle | 1 = 20 o | 2 = 30 | 3 = 45 | 4 = 60 | 5 = 90 | 6 = 108 | Quantity of angles | | | | | | | GREEN Type of Polygon: | 1 = wild | 2 = digon | 3 = triangle | 4 = quadrilateral | 5 = pentagon | 6 = hexagon | Quantity of Polygons | | | | | | | Winning When one player completes a set, the round of play is over. That player scores 10 bonus points, and all players score 1 point for each completed polygon. Play continues, through successive rounds, until one player reaches a total score of 20
John Gough — Spatial Thinking Dice Games — 2004
points. Or the player with the highest score after an agreed time wins the game.
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John Gough — Spatial Thinking Dice Games — 2004
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Polyhedron Factory Players compete to make five polyhedra. Equipment A diagram of different polyhedra target shapes: this can be compiled by students after some preparatory investigation of “polyhedra”. Colored sixsided dice: Red = vertices Blue = Edges Green = faces VerticesEdgesandFaces Tally Sheets, one for each player. Number of Players 2 or more: this may be played with a whole class, with some modification for ending the round. Playing Players take turns. In each turn a player either rolls four dice, one each of Red, Blue and Green, with the player choosing which color dice will be the extra fourth dice; or the player claims a completed polyhedron. Rolling When a player rolls, the player writes the result of the roll on his or her VEF Tally Sheet. Claiming When a player claims a completed polyhedron, the specified polyhedron is ticked, or crossed off (or is marked, as having been claimed, by placing a counter on the diagram showing that polyhedron). The player also cancels the appropriate number of edges, vertices and faces from his or her VEF Tally Sheet. Obviously player claiming a polyhedron must have enough VEF tallies in his or her Tally Sheet to match the VEF properties of the polyhedron being claimed. Scoring Each time a polyhedron is claimed, that player scores the number of faces of the polyhedron (e.g. a cube scores 6). When one player has claimed a total of five polyhedra, the round is finished, and that player scores a bonus of 5 points.
John Gough — Spatial Thinking Dice Games — 2004
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The other players end the round by pooling their remaining VEF tallies and, if possible, using this to claim one more polyhedron, and evenly sharing the score for that polyhedron. Play continues, through successive rounds, until one player reaches a total score of 50 points. Or the player with the highest score after an agreed time wins the game.
John Gough — Spatial Thinking Dice Games — 2004
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24SquareUnit Dotty Grid Game Players take turns claiming polygons, each of whose area is 24 square units. When the playing area is full, the player who scores the longest total perimeter wins the whole game. Equipment 12x12 square dotty grid paper, or geoboard. Pencil and paper for scoring. Playing Players take turns. In each turn a player draws (or makes) the outline of a polygon on any suitable empty section of the 12x12 grid, with the polygon having an area of 24 square units. The player scores the perimeter of the polygon (measured in unit lengths: if a diagonal length is included in the polygon, this length — probably a square root of some kind, such as √2 or other surd — will be approximated by the next lowest whole number. Play continues until no more room remains for claiming a polygon with area of 24 square units. The winner of the round is the player with the highest total score. Play continues, through successive rounds, until one player reaches a total score of 100 points. Or the player with the highest score after an agreed time wins the game. Variants Make the game simpler by using a smaller size area for each claimed polygon, for example, try 12 square units. Use isometric dottypaper or isometric grid paper, and play a version of the game using equilateral triangles as the building blocks for claim areas of, say, 12 triangleunits, scoring the perimeter of the claimed shape.
John Gough — Spatial Thinking Dice Games — 2004
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Pick’s Game This game is based on the mathematical theorem known as Pick’s Rule. (Try a Google websearch for more information on this, or John L. Sullivan 1973). Pick’s Rule says that any polygon that is drawn in a lattice of squaregrid dots (like the nails in a geoboard) so that its vertices (corner points) are grid or lattice points (nails), has an area (measured in square units, with the unit being the grid distance from one grid point to the next, horizontally or vertically): A = (b / 2) + i – 1, where b is the number of boundary gridpoints (that is, the number of lattice points on the sides or edges of the polygon), and i is the number of interior gridpoints lying inside the polygon. Players take turns to roll colorcoded dice, and use the results to claim polygonal areas on a dotty grid board. Equipment Use 12x12 square grid dotty paper, or a geoboard. Pencil and paper for scoring. Three ordinary dice, colored: Red = sidelengths Blue = corner dots (vertices) Green = inner dots Tally Sheet: for recording the player’s current: Number of: Boundary Lengths 1 2 3 4 5 6 Corner dots Inner dots Playing Players take turns. In each turn a player either rolls the three dice, or claims a polygonal region. When a player decides to roll the dice, the player adds the dice results to his or her current tally of boundary lengths, corner dots and inner dots. When a player claims a polygonal region the player draws (or makes on the geoboard) the intended region. This polygon must not use more or different boundary lengths, corner dots, or inner dots than the player currently has currently recorded in his or her Tally Sheet. The polygon may share part or whole of a side with another polygon, already claimed, but must not overlap or share any part of an existing polygon’s area.
John Gough — Spatial Thinking Dice Games — 2004
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Side of polygons must be “orthogonal” (that is, horizontal or vertical): no slanting lines are allowed. The player who correctly claims a polygon scores the total number of corner, boundary, and inner points of the region. Play continues until no more polygons can be claimed in the 12x12 grid board. Variants Allow slanting lines, and use two dice: Blue for edge and corner dots, Green for inner dots.
John Gough — Spatial Thinking Dice Games — 2004
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It’s a Wrap! Players compete, using the results of colorcoded dice, to make TWO mathematically different rectangular boxes, each having a correct number of faces, with sides of specific length, scoring the surface area of each completed box. For example, a 2x4x5 box has six faces; it uses four edges of length 2, four edges of length 4, and four edges of length 5. (It scores either the surface area — 76 square units: or the volume — 40 cubic units Equipment A Red dice (for lengths of sides or faceedges), a Blue dice (for number of faces), and Green dice (for number of edges). A FaceEdgeLength Tally Sheet Length | 1 | 2 | 3 | 4 | 5 | 6 | | | | | | | | | | No. of Edges | | | | No. Faces | | Pencil and paper for boxsketching, and scoring. Number of Players 2 or more. A whole class, or smaller group, can play, simultaneously, all using the successive dice rolls, but using plain dice, and deciding which of the three dice results to tally as Face, Edge or Length. Playing In each turn a player may roll, or “buy”, or declare a complete box has been “made and wrapped”. Rolling The player rolls the three colored dice and adds the results to his or her FaceEdgeLength Tally Sheet. “Buying” The player may request any of the other players ONE of the other players Edges, or Lengths, or Faces. If the other player agrees, the Tally Sheets are adjusted (one tally mark being removed, and another added), and the other player receives a bonus score of 1 extra
John Gough — Spatial Thinking Dice Games — 2004
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point for this round. Claiming a Complete Box Has Been “Made and Wrapped” The player declares the making and wrapping of a box, stating the length by width by height measurements of the claimed box, and demonstrating that he or she has the necessary correct number of faces (6, for a rectangular cuboid), and edges, as well as an appropriate number of lengths for the necessary edges. Winning Play continues until one player has correctly made TWO mathematically different boxes. (Note that a 2x5x3 box, sitting flat on a 2x5 base is mathematically identical to a 2x3x5 box “standing” upright on a smaller 2x3 base). The player who is first to correctly complete TWO boxes scores the surface area of both boxes. Each of the other players can score the surface area of any boxes they have already claimed, and they may each finish the round by making ONE box (if possible), scoring only the volume of that final box. That completes the current round of play. All Tally Sheets are erased, and play resumes with a new round. Play continues, through successive rounds, until one player reaches a total score of 500 points. Or the player with the highest score after an agreed time wins the game. HideAway Players compete, successively placing unitcubes on gridcoordinate points on a 6x6 grid board, scoring the number of concealed faces of the unitcube being placed. Equipment A 6x6 square grid board, labelled 1 to 6 along one side, and 1 to 6 along the other. (Each square in the grid is identified as a coordinate, as in Battleships, or streetdirectories.) Two ordinary dice. A set of unit cubes whose size matches the size of the squares on the grid board. Pencil and paper for scoring. Number of Players 2 or more: a whole class can play, using an OverHead Projector to display the gridboard and dice results. Playing Players take turns.
John Gough — Spatial Thinking Dice Games — 2004
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In each turn a player rolls both dice, and the uses result to identify two gridsquares on the board. The player chooses which of these (if possible) to be the location for placing a new unitcube on the grid (stacking neatly, if necessary). The player scores the number of faces on the unitcube that are “hidden” (cannot be seen) once the new cube has been placed. If three unitcubes have already been stacked on both gridsquares corresponding to the two dice outcomes, the player misses a turn. When no further play is possible the current round ends. Clear the gridboard of stacked unitcubes, and start a fresh round. Play continues, through successive rounds, until one player reaches a total score of 100 points. Or the player with the highest score after an agreed time wins the game.
John Gough — Spatial Thinking Dice Games — 2004
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Towers and Barns Players use dice results to collect cubes and claim a “box”, scoring the volume of the “box” plus the floorarea. For example, a box with measurements 2x3x4, sitting flat on one of the 3x4 faces, has a floorplan area of 12 unit “faces” touching the floor or table, while the volume of the box is 2x3x4 or 24: this box scores 36 points. But if the box is sitting on a 2x3 floorplan the score for the claimed box is volume (24) plus floor (6), or 30 points. Equipment An ordinary dice. A Playing Board, showing thirtysix rectangular floorplans of 1x1, 1x2, 1x3, … up to 6x6 square units. Alternatively, players may take turns in setting up a Playing Board, drawing TEN rectangular floorplans for “boxes” with a maximum possible volume of 36 unitcubes. Pencil and paper for sketching and scoring. Unit cubes for trialling and demonstrating. Dottypaper or isometric grid paper, for sketching. Counters, for claiming “boxes” on the Playing Board. Number of Players 2 or more players. Playing Players take turns. In each turn a player either rolls or claims. When a player rolls the dice, the player adds the dice result to his or her current tally; or may actually that number of unitcubes. When a player claims a box, on the Playing Board, the player places a counter on the Board, to show that this box is no longer available to be claimed; the players cancels the volume of the claimed box from his or her tally of cubes; and the player scores the volume of the claimed box, plus the floorarea of that box. Play continues until all boxes on the Playing Board have been claimed, or the player with the highest score after an agreed time wins the game.
John Gough — Spatial Thinking Dice Games — 2004
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HexaFence Players compete using the results of rolling colorcoded dice to tally edges and unitsided triangles, and build a complete “fence” around, and thereby claim, or occupy sections of an equilateral triangulargrid Playing Board, such as a rhombus of sides 6 units x6 units, or a large triangular region with sides 8 units long. Equipment An equilateral triangulargrid Playing Board, such as a rhombus of sides 6 units x6 units, or a large triangular region with sides 8 units long. (Or an isometric geoboard. Or isometric dotty or grid paper.) A Red dice, for unitedges. A Blue dice, for triangular unitsided areas. Optional: colorcoded counters, such as Centicubes, to claim unitsided triangles. Number of Players 2 or more players. Playing Players take turns. In each turn a player either rolls the two dice, or claims a section of the board, corresponding to (some of) his or her current tally of unitedges and unitsided triangles. When rolling, the player adds the dice results to his or her current tally of unitedges and unitsided triangles. When claiming, the player draws the boundary of a section of the Playing Board (or places counters in unitsided triangles), provided the player has at least that number of unitedges and unitsided triangles. Note that when more than one unittriangle is being claimed, all triangles must join by at least one whole unitedge: that is, triangles are not allowed to “join” only by touching endpoints, as in |><|. The player scores the number of unitsided triangles contained in the claimed region, plus the number of inner unitedges inside the claimed section. For example, a twotriangle lozenge shape has a boundary “fence” of 4 unitsides, and a single inner “edge”, and hence would score a total of 5 points. (Can you see that three unittriangles, joined together, have an outer “fence of 5 units, and two inner edges, giving a total score of 7?) Successive claimed regions can share the same “fence” or boundary. But the later claimant must have enough unitedges to be able to “fence” or surround the entire section being claimed, even though that section had already been “fenced” by the previous
John Gough — Spatial Thinking Dice Games — 2004
claimant, Play continues until the board has been filled, and no further unittriangles can be claimed. References Sullivan, J.L. (1973). "Polygons on a Lattice", The Arithmetic Teacher, December, pp 673–675.
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Playing With Space: Dice Games, and Others John Gough — [email protected] Introduction Most of the following games require pencil and paper, several ordinary sixsided dice of different colors, dotty or square grid paper, or a geoboard. Each game is designed to provide gamebased experiences that can help develop ideas about space, geometry, and concepts such as vertex, side, face, perimeter, area, surface area, volume, polygon, and polyhedra. Most of the games can be played with just two players. It is also useful to play together in pairs, with two or more pairs of players competing. Some games can be easily adapted for play by a whole class, with each person either playing individually, or as one of a pair. Make a Cube Players aim to score by building successive complete layers of a 3x3x3 cubic unit cube. Equipment A 3x3 square grid baseboard (the size of the squares fits the size of the cubes used in playing: for example, centicubes need a 3cm x3cm grid). An ordinary dice. Centicubes or other fittogether unit cubes, or Cuisenaire, or MAB materials. Pencil and paper for scoring. Number of Players 2 players Playing Players take turns. In each turn, a player rolls the dice. Depending on the result the player takes the appropriate piece or pieces: 1 = unit cube 4 = two unit cubes 2 = two cubes together, in a “stick” 5 = two lots of two cubes in a “stick” 3 = three cubes in a “stick” 6 = two lots of three cubes in a “stick” Then the player places the piece or pieces anywhere on the playing board, so that: — each piece fits in the 3x3 base (no overhanging) — this does not leave any unfilled holes in the 3x3x3 cube that is being built. Scoring A player scores 1 point for completing a whole layer (that is, placing one or more pieces that completes either the first or second “floor” of the 3x3x3 cube that is being built; this
John Gough — Spatial Thinking Dice Games — 2004
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may mean that some part of the piece or pieces used to complete the “floor” will protrude vertically above the “floor”.) The player who completes the third “floor” (that is, finishes the whole 3x3x3 cube) scores 5 points, minus each extra cube that protrudes above the level of the third “floor”. Play continues, through successive rounds of starting at “ground level”, and building the 3x3x3 cube, until one player reaches a total score of 20 points. Or the player with the highest score after an agreed time wins the game.
John Gough — Spatial Thinking Dice Games — 2004
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Polygon Factory Players use dice rolls and playing to get a maximum score as they make collections of polygons. Equipment Pencil and paper. Four ordinary dice: Red, Blue, Yellow and Green. A Player’s Table for each player. Number of Players 2 or more players Rules for Play Players aim to use the results of dice rolling, and other actions, to make a particular set of polygons, using “material” that is specified by the results of the dice rolling (which will be explained). Players take turns. In each turn a player may do ONE of the following: • roll some or all of the dice; • request to “buy” some of the “material” belonging to another player; • request to “swap”, in some negotiated way, some of his or her “material” for some of another player’s “material”; or • declare that one of a set of polygons, or a complete set of polygons, has been completed. Red dice: the result (1 to 6) specifies the QUANTITY of ONE of the other dice. Blue dice: the result (1 to 6) specifies the LENGTH of a side for a possible polygon. Yellow dice: the result (1 to 6) specifies the ANGLEsize: 1 = 20o, 2 = 30 o, 3 = 45 o, 4 = 60 o, 5 = 90 o, and 6 = 108 o. Green dice: the result (1 to 6) specifies the POLYGONtype (see the Player’s Table, below). Rolling When a player rolls the dice, the player chooses which of ONE the other dice will be selected to be combined with the Red Quantity. For example, a player might roll and obtain the following result: Red = 4, Blue = 2, Yellow = 5, and Green = 2. Hence the player can specify, that is, tally, in his or her Player’s Table: — four sides of length 2 units; or
John Gough — Spatial Thinking Dice Games — 2004
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— four angles of 90 degrees; or — four polygons which are “digons”, that is, twosided figures, otherwise known as “angles”. If a player rolls the dice and gets 1 for Green (Type of Polygon), this result is “wild”, and the player can freely use this as though it were any other dice result (that is, the player can freely use this as any other type of polygon). “Buying” If a player “buys” tallied material from another player, the other player receives a payment or bonus score of 1, and the buyer loses 1 point of score. “Swapping” If two players agree to “swap” some their materials, then each scores a bonus of 1 point. Player’s Table Possible Polygon Sets: — one of each kind: e.g. a family of triangles consists of all acuteangle scalene (three different lengths, and three different acute angles), one obtuseangle scalene, isosceles nonright, rightisosceles, equilateral. — a family of one kind: e.g. a family of angles consists of acute, right, obtuse, and reflex. BLUE Length of Side | 1 | 2 | 3 | 4 | 5 | 6 | Quantity of length | | | | | | | YELLOW o o o o o Size of Angle | 1 = 20 o | 2 = 30 | 3 = 45 | 4 = 60 | 5 = 90 | 6 = 108 | Quantity of angles | | | | | | | GREEN Type of Polygon: | 1 = wild | 2 = digon | 3 = triangle | 4 = quadrilateral | 5 = pentagon | 6 = hexagon | Quantity of Polygons | | | | | | | Winning When one player completes a set, the round of play is over. That player scores 10 bonus points, and all players score 1 point for each completed polygon. Play continues, through successive rounds, until one player reaches a total score of 20
John Gough — Spatial Thinking Dice Games — 2004
points. Or the player with the highest score after an agreed time wins the game.
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John Gough — Spatial Thinking Dice Games — 2004
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Polyhedron Factory Players compete to make five polyhedra. Equipment A diagram of different polyhedra target shapes: this can be compiled by students after some preparatory investigation of “polyhedra”. Colored sixsided dice: Red = vertices Blue = Edges Green = faces VerticesEdgesandFaces Tally Sheets, one for each player. Number of Players 2 or more: this may be played with a whole class, with some modification for ending the round. Playing Players take turns. In each turn a player either rolls four dice, one each of Red, Blue and Green, with the player choosing which color dice will be the extra fourth dice; or the player claims a completed polyhedron. Rolling When a player rolls, the player writes the result of the roll on his or her VEF Tally Sheet. Claiming When a player claims a completed polyhedron, the specified polyhedron is ticked, or crossed off (or is marked, as having been claimed, by placing a counter on the diagram showing that polyhedron). The player also cancels the appropriate number of edges, vertices and faces from his or her VEF Tally Sheet. Obviously player claiming a polyhedron must have enough VEF tallies in his or her Tally Sheet to match the VEF properties of the polyhedron being claimed. Scoring Each time a polyhedron is claimed, that player scores the number of faces of the polyhedron (e.g. a cube scores 6). When one player has claimed a total of five polyhedra, the round is finished, and that player scores a bonus of 5 points.
John Gough — Spatial Thinking Dice Games — 2004
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The other players end the round by pooling their remaining VEF tallies and, if possible, using this to claim one more polyhedron, and evenly sharing the score for that polyhedron. Play continues, through successive rounds, until one player reaches a total score of 50 points. Or the player with the highest score after an agreed time wins the game.
John Gough — Spatial Thinking Dice Games — 2004
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24SquareUnit Dotty Grid Game Players take turns claiming polygons, each of whose area is 24 square units. When the playing area is full, the player who scores the longest total perimeter wins the whole game. Equipment 12x12 square dotty grid paper, or geoboard. Pencil and paper for scoring. Playing Players take turns. In each turn a player draws (or makes) the outline of a polygon on any suitable empty section of the 12x12 grid, with the polygon having an area of 24 square units. The player scores the perimeter of the polygon (measured in unit lengths: if a diagonal length is included in the polygon, this length — probably a square root of some kind, such as √2 or other surd — will be approximated by the next lowest whole number. Play continues until no more room remains for claiming a polygon with area of 24 square units. The winner of the round is the player with the highest total score. Play continues, through successive rounds, until one player reaches a total score of 100 points. Or the player with the highest score after an agreed time wins the game. Variants Make the game simpler by using a smaller size area for each claimed polygon, for example, try 12 square units. Use isometric dottypaper or isometric grid paper, and play a version of the game using equilateral triangles as the building blocks for claim areas of, say, 12 triangleunits, scoring the perimeter of the claimed shape.
John Gough — Spatial Thinking Dice Games — 2004
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Pick’s Game This game is based on the mathematical theorem known as Pick’s Rule. (Try a Google websearch for more information on this, or John L. Sullivan 1973). Pick’s Rule says that any polygon that is drawn in a lattice of squaregrid dots (like the nails in a geoboard) so that its vertices (corner points) are grid or lattice points (nails), has an area (measured in square units, with the unit being the grid distance from one grid point to the next, horizontally or vertically): A = (b / 2) + i – 1, where b is the number of boundary gridpoints (that is, the number of lattice points on the sides or edges of the polygon), and i is the number of interior gridpoints lying inside the polygon. Players take turns to roll colorcoded dice, and use the results to claim polygonal areas on a dotty grid board. Equipment Use 12x12 square grid dotty paper, or a geoboard. Pencil and paper for scoring. Three ordinary dice, colored: Red = sidelengths Blue = corner dots (vertices) Green = inner dots Tally Sheet: for recording the player’s current: Number of: Boundary Lengths 1 2 3 4 5 6 Corner dots Inner dots Playing Players take turns. In each turn a player either rolls the three dice, or claims a polygonal region. When a player decides to roll the dice, the player adds the dice results to his or her current tally of boundary lengths, corner dots and inner dots. When a player claims a polygonal region the player draws (or makes on the geoboard) the intended region. This polygon must not use more or different boundary lengths, corner dots, or inner dots than the player currently has currently recorded in his or her Tally Sheet. The polygon may share part or whole of a side with another polygon, already claimed, but must not overlap or share any part of an existing polygon’s area.
John Gough — Spatial Thinking Dice Games — 2004
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Side of polygons must be “orthogonal” (that is, horizontal or vertical): no slanting lines are allowed. The player who correctly claims a polygon scores the total number of corner, boundary, and inner points of the region. Play continues until no more polygons can be claimed in the 12x12 grid board. Variants Allow slanting lines, and use two dice: Blue for edge and corner dots, Green for inner dots.
John Gough — Spatial Thinking Dice Games — 2004
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It’s a Wrap! Players compete, using the results of colorcoded dice, to make TWO mathematically different rectangular boxes, each having a correct number of faces, with sides of specific length, scoring the surface area of each completed box. For example, a 2x4x5 box has six faces; it uses four edges of length 2, four edges of length 4, and four edges of length 5. (It scores either the surface area — 76 square units: or the volume — 40 cubic units Equipment A Red dice (for lengths of sides or faceedges), a Blue dice (for number of faces), and Green dice (for number of edges). A FaceEdgeLength Tally Sheet Length | 1 | 2 | 3 | 4 | 5 | 6 | | | | | | | | | | No. of Edges | | | | No. Faces | | Pencil and paper for boxsketching, and scoring. Number of Players 2 or more. A whole class, or smaller group, can play, simultaneously, all using the successive dice rolls, but using plain dice, and deciding which of the three dice results to tally as Face, Edge or Length. Playing In each turn a player may roll, or “buy”, or declare a complete box has been “made and wrapped”. Rolling The player rolls the three colored dice and adds the results to his or her FaceEdgeLength Tally Sheet. “Buying” The player may request any of the other players ONE of the other players Edges, or Lengths, or Faces. If the other player agrees, the Tally Sheets are adjusted (one tally mark being removed, and another added), and the other player receives a bonus score of 1 extra
John Gough — Spatial Thinking Dice Games — 2004
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point for this round. Claiming a Complete Box Has Been “Made and Wrapped” The player declares the making and wrapping of a box, stating the length by width by height measurements of the claimed box, and demonstrating that he or she has the necessary correct number of faces (6, for a rectangular cuboid), and edges, as well as an appropriate number of lengths for the necessary edges. Winning Play continues until one player has correctly made TWO mathematically different boxes. (Note that a 2x5x3 box, sitting flat on a 2x5 base is mathematically identical to a 2x3x5 box “standing” upright on a smaller 2x3 base). The player who is first to correctly complete TWO boxes scores the surface area of both boxes. Each of the other players can score the surface area of any boxes they have already claimed, and they may each finish the round by making ONE box (if possible), scoring only the volume of that final box. That completes the current round of play. All Tally Sheets are erased, and play resumes with a new round. Play continues, through successive rounds, until one player reaches a total score of 500 points. Or the player with the highest score after an agreed time wins the game. HideAway Players compete, successively placing unitcubes on gridcoordinate points on a 6x6 grid board, scoring the number of concealed faces of the unitcube being placed. Equipment A 6x6 square grid board, labelled 1 to 6 along one side, and 1 to 6 along the other. (Each square in the grid is identified as a coordinate, as in Battleships, or streetdirectories.) Two ordinary dice. A set of unit cubes whose size matches the size of the squares on the grid board. Pencil and paper for scoring. Number of Players 2 or more: a whole class can play, using an OverHead Projector to display the gridboard and dice results. Playing Players take turns.
John Gough — Spatial Thinking Dice Games — 2004
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In each turn a player rolls both dice, and the uses result to identify two gridsquares on the board. The player chooses which of these (if possible) to be the location for placing a new unitcube on the grid (stacking neatly, if necessary). The player scores the number of faces on the unitcube that are “hidden” (cannot be seen) once the new cube has been placed. If three unitcubes have already been stacked on both gridsquares corresponding to the two dice outcomes, the player misses a turn. When no further play is possible the current round ends. Clear the gridboard of stacked unitcubes, and start a fresh round. Play continues, through successive rounds, until one player reaches a total score of 100 points. Or the player with the highest score after an agreed time wins the game.
John Gough — Spatial Thinking Dice Games — 2004
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Towers and Barns Players use dice results to collect cubes and claim a “box”, scoring the volume of the “box” plus the floorarea. For example, a box with measurements 2x3x4, sitting flat on one of the 3x4 faces, has a floorplan area of 12 unit “faces” touching the floor or table, while the volume of the box is 2x3x4 or 24: this box scores 36 points. But if the box is sitting on a 2x3 floorplan the score for the claimed box is volume (24) plus floor (6), or 30 points. Equipment An ordinary dice. A Playing Board, showing thirtysix rectangular floorplans of 1x1, 1x2, 1x3, … up to 6x6 square units. Alternatively, players may take turns in setting up a Playing Board, drawing TEN rectangular floorplans for “boxes” with a maximum possible volume of 36 unitcubes. Pencil and paper for sketching and scoring. Unit cubes for trialling and demonstrating. Dottypaper or isometric grid paper, for sketching. Counters, for claiming “boxes” on the Playing Board. Number of Players 2 or more players. Playing Players take turns. In each turn a player either rolls or claims. When a player rolls the dice, the player adds the dice result to his or her current tally; or may actually that number of unitcubes. When a player claims a box, on the Playing Board, the player places a counter on the Board, to show that this box is no longer available to be claimed; the players cancels the volume of the claimed box from his or her tally of cubes; and the player scores the volume of the claimed box, plus the floorarea of that box. Play continues until all boxes on the Playing Board have been claimed, or the player with the highest score after an agreed time wins the game.
John Gough — Spatial Thinking Dice Games — 2004
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HexaFence Players compete using the results of rolling colorcoded dice to tally edges and unitsided triangles, and build a complete “fence” around, and thereby claim, or occupy sections of an equilateral triangulargrid Playing Board, such as a rhombus of sides 6 units x6 units, or a large triangular region with sides 8 units long. Equipment An equilateral triangulargrid Playing Board, such as a rhombus of sides 6 units x6 units, or a large triangular region with sides 8 units long. (Or an isometric geoboard. Or isometric dotty or grid paper.) A Red dice, for unitedges. A Blue dice, for triangular unitsided areas. Optional: colorcoded counters, such as Centicubes, to claim unitsided triangles. Number of Players 2 or more players. Playing Players take turns. In each turn a player either rolls the two dice, or claims a section of the board, corresponding to (some of) his or her current tally of unitedges and unitsided triangles. When rolling, the player adds the dice results to his or her current tally of unitedges and unitsided triangles. When claiming, the player draws the boundary of a section of the Playing Board (or places counters in unitsided triangles), provided the player has at least that number of unitedges and unitsided triangles. Note that when more than one unittriangle is being claimed, all triangles must join by at least one whole unitedge: that is, triangles are not allowed to “join” only by touching endpoints, as in |><|. The player scores the number of unitsided triangles contained in the claimed region, plus the number of inner unitedges inside the claimed section. For example, a twotriangle lozenge shape has a boundary “fence” of 4 unitsides, and a single inner “edge”, and hence would score a total of 5 points. (Can you see that three unittriangles, joined together, have an outer “fence of 5 units, and two inner edges, giving a total score of 7?) Successive claimed regions can share the same “fence” or boundary. But the later claimant must have enough unitedges to be able to “fence” or surround the entire section being claimed, even though that section had already been “fenced” by the previous
John Gough — Spatial Thinking Dice Games — 2004
claimant, Play continues until the board has been filled, and no further unittriangles can be claimed. References Sullivan, J.L. (1973). "Polygons on a Lattice", The Arithmetic Teacher, December, pp 673–675.
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