Hybrid DI-LR Sets: Notes & Examples

BEGINNER

Hybrid DI-LR Sets: visual worked lesson

Core idea

A hybrid set combines numerical data with logical clues. Solve small restrictions first, then use them in the calculations.

Solving method

Make one master table, then use numerical and logical clues together. Recheck totals after each forced placement.

Worked example

Two rooms have 10 seats each. Room A has 8 people and room B must have two fewer. How many people are in B? 1. Use the comparison clue: B has two fewer than A. 2. 8 − 2 = 6. 3. Check capacity: 6 fits within 10 seats.

Read the diagram

The bars represent people rather than seats. The condition “two fewer than A” gives B = 8 − 2 = 6. Check 6 against the separate capacity of 10. This is the two-layer method: first interpret the relation, then calculate and verify that the result respects all limits.

Exam-style application

A,B,C deliver 10,20,30 parcels, one count each. They travel by bike, bus or van, one each. A does not use the van and delivers more than the bus user. B uses the bike and delivers 20. The bus user cannot be A or B, so C uses the bus. A must use the van, contradicting the first clue. The correct conclusion is that the clues are inconsistent, not an invented assignment. Changing "A does not use the van" to "A does not use the bike" would make A=van,30 and C=bus,10 consistent.

Common mistakes

A numerical answer must also satisfy every logical restriction. A contradiction is useful information. Do not silently drop a clue to force a table to work.

Combine a logical condition with capacity. Room B = 8 − 2 = 6; both rooms stay within capacity. — Combine a logical condition with capacity

Room B = 8 − 2 = 6; both rooms stay within capacity. — Combine a logical condition with capacity

Source record: Bharatiya Pathshala — original illustrated lessons.

BEGINNER

Core idea

A hybrid set mixes numbers with logical rules. You may need to find who owns an item and how many items each person owns. Solve simple relationships first, then calculate. A neat table prevents you from losing facts while working.

BEGINNER

Steps

1. Separate the numbers from the placement rules. 2. Fill facts that are certain. 3. Turn totals into equations. 4. Use the logical rules to reduce possible values. 5. Check the final table against both the numerical and logical conditions.

BEGINNER

Worked example

Three boxes hold 12 pencils in total. A has twice as many as B. C has 3 pencils. Then A and B together have 9. If B has x, A has 2x, so 3x = 9. B has 3 and A has 6.

BEGINNER

Common mistake

Do not guess a number that fits one sentence and forget the total. Every entry must satisfy all rules at once.

Logical Puzzles

List variables and possible values, then propagate each restriction through a grid. Branch only when deduction stops.

Worked example

Three people use three different colours. A cannot use red or blue, so A uses green. Remove green from the other two possibilities before testing the next clue.

Application and strategy

Make one master table, then use numerical and logical clues together. Recheck totals after each forced placement.

Worked application

A,B,C deliver 10,20,30 parcels, one count each. They travel by bike, bus or van, one each. A does not use the van and delivers more than the bus user. B uses the bike and delivers 20. The bus user cannot be A or B, so C uses the bus. A must use the van, contradicting the first clue. The correct conclusion is that the clues are inconsistent, not an invented assignment. Changing "A does not use the van" to "A does not use the bike" would make A=van,30 and C=bus,10 consistent.

Common traps and strategy

A contradiction is useful information. Do not silently drop a clue to force a table to work.

Combining logical constraints and numerical data

Core idea

Hybrid sets require deductions about structure before arithmetic becomes possible. Keep a table of direct facts, a separate list of deductions, and clearly labelled alternative cases.

Worked example

Two shops sell 60 items in total. A sells twice as many as B, and half A's items are notebooks. Let B=x, so A=2x and 3x=60. Then B=20, A=40, and A sells 20 notebooks. None of this determines how many notebooks B sells without another clue.

Solving method

Use totals, capacity bounds, parity and ratios to restrict cases. For a sports table, use the supplied scoring and qualification rules rather than assuming them. If each match awards 3 points for a win and 1 per team for a draw, the total points over M matches equal 3M−D, where D is the number of draws.

Common mistakes

A case valid under one branch is not necessarily forced. Never combine cells from incompatible branches into one table, or use an answer that depends on a missing chart.

Source record: Bharatiya Pathshala — reviewed preparation lessons.

Mathematical puzzles and ambiguity

Core idea

Translate a puzzle into integer, divisibility or ordering conditions. Enumerate small cases systematically, then ask which additional clue distinguishes them.

Worked example

Three children have positive integer ages with product 36. The unordered triples and sums are: (1,1,36):38; (1,2,18):21; (1,3,12):16; (1,4,9):14; (1,6,6):13; (2,2,9):13; (2,3,6):11; (3,3,4):10. If an observer knows the sum but still cannot determine the ages, the sum must be 13. A further clue that there is a unique oldest child selects (2,2,9), rather than (1,6,6).

Common mistakes

The conclusion depends on positive integer ages and on what the observer knows. Do not treat a vague reference to “the oldest” as a unique-age clue unless the intended puzzle convention supports it. Factor and parity checks narrow the search but do not replace testing all conditions.

Source record: Bharatiya Pathshala — reviewed preparation lessons.

Solving assignment puzzles by elimination

Core idea

An assignment puzzle matches entities to attributes, often one-to-one. Write each domain and whether values may repeat. Use a grid to record allowed and forbidden assignments.

Worked example

A, B and C each choose a different colour from red, blue and green. A cannot choose red or blue, so A chooses green. B cannot choose blue, and green is already used, so B chooses red. C then chooses blue. Every assignment follows from an explicit exclusion and the one-to-one condition.

Solving method

Place fixed values first, propagate exclusions across rows and columns, and branch on an unresolved value only when deduction stops. If a branch violates a rule, discard the whole branch. Distinguish “could be true” from “must be true” by comparing all surviving cases.

Common mistakes

Do not assume all attributes are different unless the problem says so. An incomplete example without its original grid cannot justify a unique final assignment.

Source record: Bharatiya Pathshala — reviewed preparation lessons.

Analytical reasoning: necessary and possible conclusions

Core idea

A necessary conclusion holds in every arrangement satisfying the rules. A possible conclusion needs just one valid arrangement. An impossible conclusion holds in none.

Worked example

Three tasks A,B,C occur once each, with A before C. The valid orders are ABC, ACB and BAC. “C is last” is possible but not necessary. “C precedes A” is impossible. “A is not last” is necessary. The position of B cannot be fixed from the single rule.

Solving method

Represent each rule precisely, including “only if”, “unless” and “exactly one”. Seek a counterexample to a claimed necessity, or a complete valid construction to establish possibility. Local consistency is insufficient if another rule is violated elsewhere.

Common mistakes

Never combine totals from different hypothetical cases. A worked answer requires the original constraints; isolated solution tables or answer numbers do not establish a result.

Source record: Bharatiya Pathshala — reviewed preparation lessons.

Hybrid DI-LR Sets, explained simply

Try a small example

Core idea

Steps

Worked example

Common mistake

Logical Puzzles

Worked example

CAT: Data Interpretation and Logical Reasoning

Hybrid DI-LR Sets, explained simply

Try a small example

Hybrid DI-LR Sets: visual worked lesson

Core idea

Solving method

Worked example

Read the diagram

Exam-style application

Common mistakes

Core idea

Steps

Worked example

Common mistake

Logical Puzzles

Worked example

Application and strategy

Worked application

Common traps and strategy

Combining logical constraints and numerical data

Core idea

Worked example

Solving method

Common mistakes

Mathematical puzzles and ambiguity

Core idea

Worked example

Common mistakes

Solving assignment puzzles by elimination

Core idea

Worked example

Solving method

Common mistakes

Analytical reasoning: necessary and possible conclusions

Core idea

Worked example

Solving method

Common mistakes

CAT: Data Interpretation and Logical Reasoning

Hybrid DI-LR Sets: visual worked lesson

Core idea

Solving method

Worked example

Read the diagram

Exam-style application

Common mistakes

Application and strategy

Worked application

Common traps and strategy

Combining logical constraints and numerical data

Core idea

Worked example

Solving method

Common mistakes

Mathematical puzzles and ambiguity

Core idea

Worked example

Common mistakes

Solving assignment puzzles by elimination

Core idea

Worked example

Solving method

Common mistakes

Analytical reasoning: necessary and possible conclusions

Core idea

Worked example

Solving method

Common mistakes