The Fruit-Basket Matrix
A 3×3 matrix of picture cards is laid out. Each card shows one fruit in one basket. The completed cards are described row by row. Row 1: apple in a small basket; apple in a medium basket; apple in a large basket. Row 2: banana in a medium basket; banana in a large basket; banana in a small basket. Row 3: cherry in a large basket; cherry in a small basket; [missing card]. Each card also has a leaf on the fruit: in row 1 the leaf points up; in row 2 it points right; in row 3 it points down. Every row contains one small, one medium, and one large basket. Every column also contains one small, one medium, and one large basket. A student proposes the missing card is “cherry in a large basket with leaf pointing down” because the diagonal from top-left to bottom-right has large baskets. Another student proposes “cherry in a medium basket with leaf pointing down” because each row and column must contain all three sizes. The coach asks for the most defensible answer, the rule that supports it, and a reason the other proposal is weaker.
1. What is the missing card, and which two independent constraints fix it?
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Missing card: cherry in a medium basket, leaf pointing down. Row 3 already has a large and a small basket, so the row needs medium. Column 3 already has a large basket in row 1 and a small basket in row 2, so the column also needs medium. Those are two independent constraints pointing to the same size. The fruit is fixed by the row: row 3 is the cherry row. The leaf direction is fixed by the row pattern: row 1 up, row 2 right, row 3 down. Thus the card is cherry–medium–down.
2. Evaluate the first student’s diagonal argument. Is the diagonal actually all large baskets?
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No. The diagonal from top-left to bottom-right runs small (row 1, column 1), large (row 2, column 2), and the missing cell-so it is not an all-large diagonal. The all-large diagonal is the other one: row 1 column 3, row 2 column 2, row 3 column 1. Even if the student had named the correct diagonal, a diagonal pattern would not override the row-and-column size rule. The diagonal is a by-product of the Latin-square arrangement, not a separate law.
3. Which rule is stronger, and what observation would distinguish the two proposed rules?
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The row-and-column rule is stronger because it applies to every cell and predicts the missing size from two directions. The diagonal rule covers only three cells and is underdetermined: it does not explain why row 3 has a large and a small basket already. A distinguishing observation would be a fourth row or a changed column order. For example, if the matrix were extended with a new row and the diagonal still held but a column repeated a size, the diagonal rule would survive while the Latin-square rule would fail.