Rule Learning

What you see

Each card shows one to three copies of a shape. Cards vary on up to four attributes:

• shape: circle, square, triangle or diamond
• fill: outline, striped or solid
• count: 1, 2 or 3
• size: small or large

At low levels only some attributes vary; the others are fixed (single, small, outline) so they cannot matter.

The prompt asks “Which rule puts a card in Belongs?” The evidence sits in two framed groups, Belongs and Doesn't belong, drawn the same way. Under them is the rule builder, where you declare the rule:

1. In the Form row, choose Is (one test holds), Is not (the test fails), And (both tests hold) or Or (either holds). Only the forms your level allows are offered, and the row appears only on levels with more than one form.
2. In the Test row, choose an attribute chip (Shape, Fill, Count or Size, as the level uses them) and then a value chip. Shapes and fills are shown as small glyphs, size as a small or large circle, and count as 1, 2 or 3. From level 8 the count values also include ≥ 2 and ≥ 3, meaning at least 2 and at least 3.
3. For And and Or the builder shows Test 1 and Test 2; fill the second the same way.

As you build, the line under the board reads your rule back as a sentence, for example “Belongs when the fill is solid and the count is 2.” It only shows what your rule says; it does not say whether you are right. Number keys choose chips in the row being filled, counting from the left, and the Clear button (key C) under the builder starts the rule again. Press Check answer (or Enter) when you are done. Any equivalent way of writing the rule counts. After a wrong answer the same line shows the hidden rule, for example “The hidden rule: the shape is a circle or the count is 3.”

How the evidence is chosen

This is the most useful thing to know about the game. The generator does not show random cards. It lists every rule that can be built at your level, then picks evidence cards one at a time, each time choosing the card that splits the surviving rules most evenly, until only the hidden rule (or rules that sort every possible card the same way) is left. It may then add a few extra cards to reach the level's minimum.

So nearly every card is there for a reason. If a card seems pointless, check whether it is the one that kills the rule you are about to declare.

What changes as levels rise

There is one more constraint that shapes what you will actually meet. The generator only uses hidden rules that cover between 20 and 80 per cent of all possible cards. An And of two exact values is too narrow for that (triangle and solid covers one card in twelve), so:

• Up to level 5, the hidden rule is always a single test, even though And is offered in the builder.
• From level 6, most hidden rules are Or rules: about four in five of the rules the generator can use at level 6, and roughly three in four at levels 7 to 9.
• And rules appear only from level 8, and always pair a count of at least 2 (≥ 2) with a fill or a size.
• At level 10 there is no plain Is, but Is not is still available, so single-attribute rules can still appear in negated form.

What it measures

Rule Learning measures inductive reasoning: inferring a general rule from labelled examples. It feeds the rule learning facet of your reasoning score, carrying 13 per cent of the dimension. There is a deductive component too: once you have a candidate rule, you check it against every card, which is strict logic.

The task descends from the concept-attainment experiments of Bruner, Goodnow and Austin, published in A Study of Thinking. Their participants worked through cards that varied on several attributes, learned which ones were examples of a concept, and had to state it. Bruner and colleagues described concept attainment as an active search for, and testing of, the attributes that separate examples from non-examples, carried out with deliberate strategies rather than passive absorption.

Later work established which rules are hard:

• Shepard, Hovland and Jenkins found that classifications based on a single attribute are learned fastest, and that an exclusive-or over two attributes is clearly harder.
• Neisser and Weene ordered logical concepts into levels: single attributes at the bottom, conjunctions and disjunctions in the middle, and biconditionals and exclusive-or at the top.
• Feldman showed, across 41 kinds of logical concept, that difficulty is proportional to Boolean complexity: the length of the shortest formula that describes the concept. He also noted the long-standing finding that AND concepts are learned more easily than OR concepts.

That last point matters here. From level 6 the game mostly uses Or rules, the form people find less natural. If you default to looking for "what all the belonging cards share", you will miss rules where the belonging cards share one of two things.

How it is scored here

Your declared rule is compared with the hidden one over every card that could exist at your level, not just the evidence.

• If the two rules sort every possible card the same way, the trial is Correct: full credit. “The fill is not outline” and “the fill is striped or the fill is solid” are the same rule, and either counts.
• Otherwise it is a near miss, scored by agreement and marked Partly correct, or Incorrect when it scores zero. The app counts the share of all possible cards your rule sorts correctly. At 50 per cent agreement or below, the trial scores zero; above that, credit rises in a straight line, reaching just under half at almost 100 per cent. A near miss can never earn more than half of an exact rule. Its verdict reads, for example, “Sorts 75% of possible cards as the hidden rule does”.

As in the other reasoning games, 85 per cent of each trial's value comes from this accuracy and 15 per cent from speed relative to the expected time. Difficulty for this game is taken from the level. You start at level 3; an exact rule moves the next trial up, a near miss keeps the level where it is, and a rule that earns nothing moves it down.

Your profile shows the share of trials where you found the exact rule, and your typical time to settle on a rule. The session report for a near miss shows the hidden rule and how much of the card space your rule agreed with.

Where you will meet it

Rule discovery from labelled examples sits under classification and coding items in written tests, and under inductive-logical games in game-based ones.

AMCAT's Logical Ability module lists classification and coding pattern recognition among its inductive topics, and letter-coding items in most Indian placement tests are the same move in text: infer the rule from a few coded examples, then apply it. Rule Learning makes you state the rule explicitly, which is harder than choosing an option and a good way to find out whether you really had it.

The method

1. Read the Belongs group, attribute by attribute

   For each attribute, list the values that appear among the belonging cards. An attribute with a single value there (all striped, say) is a strong lead.

2. Try every single test first

   For each lead, check the Doesn't belong group. If no non-belonging card has that value, you have found the rule. Test all attributes, not just the first that looks promising.

3. If no single test works, try Or next

   Ask: which two values between them cover every belonging card? Then check that no non-belonging card has either value. From level 6 this is where most rules live.

4. Then Is not, then And

   If the Doesn't belong group is the uniform one (all its cards share one value), the rule probably says the attribute is not that value (form Is not). From level 8, if the belonging cards all have two or more shapes and share a fill or a size, try And with count ≥ 2.

5. Check every card, then read the sentence

   Run your rule over every evidence card in both groups. One exception means it is wrong. Then read the sentence under the board; if it is not the rule you meant, fix the chips before you check.

Worked example 1

Show the solution

Single tests. Belonging shapes: circle, circle, square, diamond, triangle. Not one shape. Belonging fills: outline, striped, solid, solid, solid. Not one fill. Counts and sizes vary on both sides. No single test works.

Leads. Two values stand out: two of the belonging cards are circles, and the other three are solid. Every belonging card is a circle or solid.

Check the other group. Six cards: outline squares, a striped triangle, outline diamonds, striped squares, outline triangles, a striped diamond. None is a circle, and none is solid.

Declare: form Or; Test 1 Shape, circle; Test 2 Fill, solid. The line under the board reads “Belongs when the shape is a circle or the fill is solid.”

Why the near misses fall short. "Fill is solid" alone misclassifies the 12 possible circles that are not solid, so it agrees on 60 of 72 possible cards (83 per cent) and would earn a third of full credit. "Shape is circle" alone misses the 18 possible solid cards that are not circles, so it agrees on 54 of 72 (75 per cent) and would earn a quarter.

Worked example 2

Show the solution

Single tests. All four belonging cards are large, and all have two or three shapes. Shapes and fills vary. So both "size is large" and "count is at least 2" are leads.

Check each lead alone. "Size is large" fails: the solid square and the striped triangle in the other group are large but single. "Count is at least 2" fails: several cards in the other group have two or three shapes but are small.

Combine them. Every belonging card is large and has at least two shapes. Every card in the other group fails one of the two: the large ones are single, the multiple ones are small.

Declare: form And; Test 1 Size, large; Test 2 Count, ≥ 2. (Declaring “count is 2 or count is 3” for the second part is not possible inside an And, which is why ≥ 2 exists.)

Why the near misses fall short. "Size is large" alone agrees on 60 of 72 cards (83 per cent) and would earn a third of full credit. "Count is at least 2" alone agrees on 48 of 72 (67 per cent) and would earn a sixth.

Tips and traps

Levelling up

The big steps are at level 4 (And appears in the builder, though the hidden rule is still a single test until level 8), level 6 (Or, which quickly becomes the most common hidden rule), level 7 (Is not) and level 8 (the ≥ 2 and ≥ 3 count values). At levels 9 and 10 the evidence gets thinner, so each card carries more weight, and at level 10 there is no plain Is form.

• Levels 1 to 5: build the habit of checking every attribute on both groups. The rule is always one test.
• Levels 6 to 7: after the single tests fail, go straight to Or.
• Levels 8 to 10: remember the ≥ 2 and ≥ 3 count values, and expect fewer cards, so every card matters.

1. Week 1
   • Three sessions of 10 trials at levels 3 to 5
   • For every trial, write the attribute-by-attribute check in your head before building
   • After a near miss, find the card your rule got wrong
2. Week 2
   • Push to levels 6 and 7
   • Make Or your second move whenever single tests fail
   • Count how many of your misses were And rules that should have been Or rules
3. Week 3
   • Work at levels 8 to 9 with the ≥ 2 and ≥ 3 count values
   • Alternate with Rule Sorter sessions for rule discovery without declaring
   • Aim for an exact-rule rate above 70 per cent
4. Week 4
   • Level 10 trials with thin evidence
   • Mix in ColorCode for policy learning with several rules
   • Rest before the real test

Check yourself

5 practice questions

1. How does Rule Learning decide whether your rule is correct?
2. Your rule agrees with the hidden rule on 75 per cent of possible cards but is not equivalent. Roughly what credit does it earn?
3. At level 5, the Belongs group shows only large striped cards. What kind of hidden rule is it?

and 2 more

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Sources

1. Bruner, J. S., Goodnow, J. J., & Austin, G. A. (1956). A Study of Thinking. Wiley.
2. Shepard, R. N., Hovland, C. I., & Jenkins, H. M. (1961). Learning and memorization of classifications. Psychological Monographs: General and Applied, 75(13), 1–42. doi:10.1037/h0093825
3. Neisser, U., & Weene, P. (1962). Hierarchies in concept attainment. Journal of Experimental Psychology, 64, 640–645. doi:10.1037/h0042549
4. Feldman, J. (2000). Minimization of Boolean complexity in human concept learning. Nature, 407, 630–633. doi:10.1038/35036586
5. Wason, P. C. (1960). On the failure to eliminate hypotheses in a conceptual task. Quarterly Journal of Experimental Psychology, 12(3), 129–140. doi:10.1080/17470216008416717
6. Aon. Prepare for your online assessment: logic tests. aon.com
7. SHL. Cognitive assessments: Verify Interactive Inductive Reasoning. shl.com
8. AMCAT. AMCAT syllabus: Logical Ability. myamcat.com/amcat-syllabus

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