AN ATTRIBUTE GAME
The basic game uses the basic set of attribute items. These items vary by size, color, shape. In the basic set, the sizes are large and small, the colors are blue and red, the shapes are circle and diamond.
= = t ¹ = = t t
This game also uses 6 labels: S, L, R, B, C, D. The basic game is played on a board laid out with 2 intersecting circle which are used as a Venn diagram. This board partitions the attribute items into 4 subsets each having 2 elements, depending on the values chosen for the labels, as indicated below.
Attribute Game Rules for the Basic Game: The game involves teams, with 2 to 4 people on each. To start, Team 1 selects 2 labels of different attribute types and places them face down beside the circles. Team 2 chooses any token. Team 1 must put this token in the correct place. Team 2 must place the remaining tokens in the 4 indicated sets. On each trial, Team 1 either verifies the choice or removes the token if it is in the wrong place and gives it back to Team 2. Team 1 collects one point each time they return a token, however if they return a token incorrectly or allow it to be placed incorrectly, they forfeit the game. Team 2 may try a rejected token elsewhere or try a different token. Once Team 2 has placed all the tokens they must either correctly identify the labels or forfeit the game. The goal of the game for Team 2 is to identify the labels giving Team 1 as few points as possible in the process. If they correctly identify the label then Team 1 and Team 2 reverse roles and play is repeated. If there is no forfeit the team with the most points is the winner.
For a more challenging game, use a three-circle Venn diagram. You can also use an expanded set of attribute items along with additional appropriate label
Educational Remarks: This game can be used one of several exercises in the study of strategy design concepts. It can also be used to illustrate non-numerical mathematical reasoning.
Strategy Design Problem: Recall that a correct placement does not give the opposing team a point. For the basic game, describe a strategy that will result in never giving the other team more than 3 points, no matter how much luck goes against your team. Apply the concept of expected value to you strategy.
Mathematical Reasoning: Suppose there are labels, which we cannot see, but we know that src (the small red circle) goes the left part of the left circle. We can deduce that lbd (the large blue diamond) belongs in the right part of the right circle as follows. For two items to be inside of the same circle they must share an attribute. Since src is in the left circle, lbd cannot be in the left circle. Since src is in outside of the right circle, lbd must be inside of the right circle. Thus lbd is in the right part of the right circle.
The reasoning makes no appeal to numbers yet any mathematician would agree that it was mathematical. His criteria might involve such observations as:
- We reason within a definite network of mathematical concepts. (No fair using a label of dog).
- The reasoning is general,
since similar reasoning could be applied to any token and its opposite,
regardless of where it was placed. - The reasoning is conceptual
and cannot be contradicted by paraceptual facts. (If I turn the card over
and find the large blue diamond lbd doesn’t go where I claimed then I am not really wrong because my opponent violated the rules.)