Two Rules Fit the Van Series
Each card in a five-card picture series shows a delivery van, and the series follows rules on three features at once: the border shape, the number of dots on the roof, and the shade of the van. The cards read: 1: circle border, 1 dot, light van 2: square border, 2 dots, dark van 3: triangle border, 3 dots, light van 4: circle border, 4 dots, dark van 5: square border, 5 dots, light van Two students predict card 6. Ana says: triangle border, 6 dots, dark van. Ben says: triangle border, 6 dots, light van. Both agree on the border and the dots, and neither can point to a card that contradicts the other's shading rule. The teacher asks the class to decide which shading rule the series supports.
1. Justify the border and dot predictions Ana and Ben share. State each rule and test it against all five cards.
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Border - a cycle of length 3. circle, square, triangle, then circle, square. Card 4 is the proof: the border restarts, so the rule is a three-step cycle, not a one-way run. Two steps past the second square brings card 6 to a triangle.
Dots - add one per card. 1, 2, 3, 4, 5 with no exception and no reset. Card 6 carries 6 dots.
Both rules are forced: each fits all five cards, and no rival rule fits them as simply. Ana and Ben therefore agree on two of the three features; their whole disagreement is about shading.
2. Ana's shading rule is 'light, dark, light, dark, light, so card 6 is dark.' Ben's is 'circles and triangles are light, squares are dark, so card 6 is light.' Both fit all five cards. Which is the better reading, and why?
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Why the five cards cannot decide it. The shading alternation has period 2; the border cycle has period 3. Two cycles of periods 2 and 3 stay in step for exactly their lowest common multiple, 6 cards - so cards 1–5 are guaranteed to look consistent with Ben's border-linked rule even if shading is really just alternating. The coincidence is an artefact of the periods, not evidence of a link. The two rules first disagree at card 6, which is precisely the card in question.
Why Ana's rule is the better reading
- The series is described as three separate features; shading should have its own rule. Ben's rule makes shading a copy of the border, so the third feature would carry no information at all.
- A sequence that includes a third feature usually intends it to be used; a perfectly redundant feature is a wasted attribute.
- Ana's rule is a single clause and gives three light and three dark cards in every block of six.
Honest caveat: the five cards cannot prove Ana right. Ben's rule is a claim about a hidden link between border and shade, and only card 6 or later could test it. Judgment, not deduction, picks between the two - but it is a reasoned judgment, not a guess.
3. Ben objects that the problem is unfair because 'both answers are equally correct.' Respond to him: is he right that the series cannot decide, and what would settle the question?
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Ben is half right.
Right: five cards genuinely cannot decide the question, because both rules fit every card shown. Any claim that the data prove Ana's answer would be false.
Wrong: the answers are not equally correct. When two rules fit equally well, the choice is made on simplicity and on how the sequence is constructed: shading as an independent alternating feature explains the five-card coincidence as a by-product of periods 2 and 3, whereas Ben's rule must assume an unexplained link between two attributes. That is a reason to prefer Ana's rule, though not a proof.
What would settle it: card 6 itself (the two rules disagree there), a written statement of the design rule, or a longer run of cards in which border and shading fall out of step - for example a circle card that appears in a dark position.