Grouping and Selection Puzzles

18 minintermediateModule 4: Analytical ReasoningContinues from hat-ordering-scheduling

18 min · Module 4: Analytical Reasoning · Puzzles

Why This Matters

Grouping puzzles ask you to sort people or things into teams, committees or categories while obeying conditions. Unlike a seating row there is no order to exploit, so the only visible structure is which group each item joins. That makes them feel shapeless, and candidates who try to hold the conditions in their heads lose the most time here.

Selection puzzles are the same family with a twist: not every item is used, and the question may only ask which items can be chosen. A committee of four from seven people is a grouping puzzle whose second group is simply "not selected". Master the grid and both varieties collapse into one method.

A Simple Model

Draw a grid: items down the side, groups across the top. Every condition becomes a mark or a cross, and the puzzle turns into a fill-in exercise.

List the items. Write every person or object once, down the left edge.
List the groups. Teams, committees, boxes - one column each, with the allowed size in the header.
Place the fixed items. "A is on Red" is a tick in that cell and crosses across the rest of A's row.
Translate conditionals. "If X is in, Y is out" forbids a combination - mark it and stop re-reading the sentence.
Count the slots. If a group needs two members and only two items can still go there, they are forced together.
Split only when stuck. Assume one placement, run it out, and discard the case if it breaks a rule.

The model omits the one thing experience supplies: which item to branch on first. The reliable rule of thumb is to branch on the item named in the most conditions.

The Actual Detail

Condition to grid

Condition in words	Mark on the grid
A must be on Red	Tick A-Red, cross A across the other teams
A and B are always together	One block that moves as a unit
A and B are never together	They cannot share any group
At least one of A, B is selected	Forbid the selection with neither
At most one of A, B is selected	Forbid the selection with both
If A is in, then B is in	Forbid A without B

Conditionals and their contrapositives

An implication and its contrapositive are the same statement. "If A is in, then B is in" is identical to "If B is out, then A is out". "If A is in, then B is out" is identical to "If B is in, then A is out". Spotted this way, such a condition says only one thing: these two can never both be in. Both may still be out, which is the trap.

in/out = a two-group grid · group size = number of slots · leftovers = the "not selected" column

        Red   Blue   Green
A       in     -      -
B        -     ?      ?
C        -     ?      ?
D        -     ?      ?
E       in     -      -
F        -     ?      ?
size     2     2      2

Three question forms

Must be true - a fact forced in every valid assignment; if it changes between your cases it is not forced.
Could be true - test each option against the conditions and reject the breakers; the survivor wins.
Cannot be true - the fastest, because a single broken condition kills the option.

Read the group sizes first. The sizes tell you how much freedom is left. When three teams each take two of six items, knowing one team fixes the arithmetic for the others.

"Either A or B" is ambiguous in everyday speech. In these papers it usually means at least one, and sometimes it means exactly one. If the answer hinges on it, check which reading keeps the other conditions consistent - only one usually will.

A Worked Example

Six students A, B, C, D, E and F are assigned to three teams - Red, Blue and Green - with exactly two students per team.

A is on Red.
E is on the same team as A.
D is not on the same team as A.
B and C are on the same team.
F is not on Red.

Fill the grid:

A is on Red and E joins A, so Red is full: Red = A, E.
D cannot share with A, and F is not on Red, so D and F must be in Blue and Green - the only teams with room left.
B and C occupy both slots of one team, so they take either Blue or Green entirely.
Therefore the other of Blue and Green is filled by D and F.

The placement is not unique: either Blue = B, C and Green = D, F, or the reverse. What is forced is the pairing structure - A with E, B with C, and D with F. That is the must-be-true answer the puzzle is really after.

Try It

In the worked puzzle, which statement must be true?

A condition reads "If D is selected, then F is not selected." Which statement is equivalent to it?

A committee is chosen from A, B, C, D, E with the rule "at most two of A, B, C". Which selection of four breaks the rule?

For a question asking which option "must be true", the correct method is to:

Check Your Understanding

Without looking back: how do you draw a grouping puzzle, and what does "if A is in then B is out" really forbid?

The answer: a grid with items down the side and groups across the top; and the condition forbids only the case where A and B are both in.

In the grid, items run down the side and groups run across the . "If X is in then Y is out" is equivalent to "if Y is in then X is ". "At most two of A, B, C" forbids any selection containing of them. A "could be true" question is answered by testing each option and using .

Apply It

Seven students A, B, C, D, E, F and G form a committee of exactly four. E must be on the committee. A and B cannot both be chosen. If C is chosen, D must be chosen. D and E cannot both be chosen. Who must be on the committee?

Show the reasoning

E is fixed on the committee. Since D and E cannot both be chosen, D is out. The rule "if C is chosen then D must be chosen" has contrapositive "if D is not chosen then C is not chosen", so C is out too.

That leaves seats to fill from A, B, F and G, with exactly three needed. A and B cannot both be chosen, so at most one of them can be used. To reach three from four items while taking at most one of A and B, the committee must take both F and G, plus exactly one of A or B.

So every valid committee contains E, F and G, contains exactly one of A or B, and never contains C or D. The must-be-true facts are E, F and G on the committee, and C and D off it - even though the exact identity of the fourth member is left open.

Connect

Grouping is the family of puzzles where items must be sorted. The next lesson keeps the shared-setup habit but changes the picture: a network of one-way circuits and two-way radios, where every question asks for a path and an intermediary count.