9.7 Odd-One-Out, Classification, and Logic Puzzles

9.7 Odd-One-Out, Classification, and Logic Puzzles

Classification items look casual and are not. Every well-built odd-one-out question hides exactly one property that all but one member share, and the wrong answers are built from properties that almost work. Puzzle items go further and ask you to hold several constraints in your head at once. Both reward method over speed.

The odd-one-out method

1. Read every option before theorising. The first pattern that occurs to you is usually the one the examiner planted as a decoy.
2. Name a candidate property that at least three members clearly share.
3. Count. In a set of $n$ items the defining property must hold for exactly $n-1$ of them. A property that splits a five-item set three against two is not the rule.
4. Test at least two more candidate properties before committing. If two survive and point at different members, apply the disambiguation conventions below.

Numerical classification

The divisibility shortcuts are worth memorising because they turn a slow division into a glance:

• 2 — the last digit is even.
• 3 — the digit sum is divisible by 3.
• 4 — the last two digits form a multiple of 4.
• 5 — the number ends in 0 or 5.
• 6 — it passes both the 2 test and the 3 test.
• 9 — the digit sum is divisible by 9.
• 11 — the alternating sum of the digits is divisible by 11.

Worked example. Which number does not belong: 34, 55, 68, 85, 102?

• Parity. Even members are 34, 68 and 102; odd members are 55 and 85. That is a three-against-two split, so by step 3 the property is discarded.
• Digit sums. They are 7, 10, 14, 13 and 3 — no four-against-one split.
• Common factor. 34 = 17 x 2, 68 = 17 x 4, 85 = 17 x 5 and 102 = 17 x 6, but 55 = 5 x 11. Four members are multiples of 17 and one is not.

Answer: 55. Notice that the parity check, which most candidates try first, produces nothing usable — it is the counting rule in step 3 that saves you from guessing.

Verbal classification

Figural classification

   (1) equilateral      (2) square        (3) regular          (4) circle
       triangle                              pentagon
         /\               +------+             /\               .----.
        /  \              |      |            /  \             /      \
       /    \             |      |           |    |           |        |
      /______\            +------+           \____/            \      /
                                                                '----'
   straight sides   3          4                 5                   0
   axes of symmetry 3          4                 5              unlimited
   made of          lines      lines             lines              curve

For figures, run through this fixed list of properties: number of sides or vertices, number of axes of line symmetry, order of rotational symmetry, open versus closed outline, convex versus concave (does any interior angle exceed 180 degrees?), curved versus straight edges, and shading or fill. In the row above, the odd figure is the circle: it is the only shape with no straight sides, and it is the only one whose axis count does not equal its side count.

When two answers both look right

Genuine ambiguity is common in poorly written practice material and rare in a properly edited examination. Three conventions resolve it.

1. The exactly-all-but-one rule. For guitar, violin, cello and flute, 'string instrument' holds for three of the four, so the flute is odd. A candidate might object that the violin and cello are bowed while the guitar is plucked — but 'bowed' holds for only two of four, so it cannot be the defining property.
2. The most-specific-property rule. When two properties each isolate a different member, choose the one satisfied by the tighter, more structured class. In 2, 3, 5, 7, 9, the property 'even' isolates 2 while the property 'prime' isolates 9. Being prime is a far more restrictive classification than mere parity, so the intended answer is 9, and the number 2 is the planted decoy.
3. Meaning beats spelling. Prefer a property about what the items are over an accident of how they are written, such as letter count or first letter.

Logic-grid puzzles

A logic grid matches a set of entities against two or more attributes. Build one sub-grid for each entity-attribute pairing, use a tick for a confirmed match and a cross for an exclusion, and obey three mechanical rules:

• Elimination rule. A tick in a cell forces a cross into every other cell in its row and every other cell in its column.
• Last-cell rule. If a row or a column has exactly one cell left uncrossed, that cell is a tick.
• Transfer rule. Facts move between sub-grids. If Bernard is in Room 201 and the Room 201 examinee takes Psychology, then Bernard takes Psychology.

A fully worked grid

Four applicants — Althea, Bernard, Carmela and Dario — sat the AdUCET in Rooms 201, 202, 203 and 204. Each came from a different senior high school strand (ABM, GAS, HUMSS, STEM) and each named a different target course (Accountancy, Architecture, Chemical Engineering, Psychology).

1. Bernard sat in Room 201.
2. The examinee in Room 201 named Psychology.
3. The examinee who named Chemical Engineering sat in Room 204.
4. Dario sat in a lower-numbered room than Althea, and Althea sat in a lower-numbered room than the Chemical Engineering examinee.
5. The STEM examinee sat in Room 201.
6. The HUMSS examinee named Architecture.
7. Carmela is not from the GAS strand.
8. Dario did not name Architecture.

Deduction order.

• Step 1 (clues 1 and 2, transfer rule). Bernard is in Room 201, so Bernard named Psychology. Cross Psychology out for everyone else.
• Step 2 (clues 3 and 4). The Chemical Engineering examinee is in Room 204, and clue 4 says Dario is below Althea, who is below Room 204. Neither Dario nor Althea can therefore occupy Room 204, and Bernard already holds 201. Carmela is in Room 204.
• Step 3 (clue 3, transfer rule). Carmela named Chemical Engineering.
• Step 4 (clue 4 again). Only Rooms 202 and 203 remain, and Dario is below Althea, so Dario is in 202 and Althea in 203.
• Step 5 (clue 8, last-cell rule). Accountancy and Architecture are the two courses left, for Dario and Althea. Dario is barred from Architecture, so Dario named Accountancy and Althea named Architecture.
• Step 6 (clue 5, transfer rule). The STEM examinee is in Room 201, which is Bernard, so Bernard is STEM.
• Step 7 (clue 6, transfer rule). The HUMSS examinee named Architecture, which is Althea, so Althea is HUMSS.
• Step 8 (clue 7, last-cell rule). ABM and GAS remain for Carmela and Dario, and Carmela is barred from GAS, so Carmela is ABM and Dario is GAS.

  Room grid after Step 4          Final solution
            201 202 203 204
  Althea     x   x   V   x        Althea   203  Architecture  HUMSS
  Bernard    V   x   x   x        Bernard  201  Psychology    STEM
  Carmela    x   x   x   V        Carmela  204  Chem. Eng.    ABM
  Dario      x   V   x   x        Dario    202  Accountancy   GAS

Every clue does real work here, and clue 4 is consulted twice — once to place a strand and again to fix the room. The solution is unique. If you finish a grid with a clue that never mattered, you have almost certainly made an error somewhere.

Seating arrangements

Draw the seats before you read the clues a second time, and place the absolute clues first ('at the extreme left', 'exactly in the middle of five'). Only then apply the relative clues.

Worked example. Ana, Bea, Carlo, Dina and Elmo occupy seats 1 to 5 from left to right. Carlo sits in seat 3. Elmo sits immediately to Carlo's right. Bea sits immediately to Ana's right. Dina sits at one end. From clue 2, Elmo takes seat 4. Seats 1, 2 and 5 remain, and the only adjacent pair among them is 1 and 2, so Ana takes seat 1 and Bea seat 2. Dina therefore takes seat 5, which is indeed an end. Order: Ana, Bea, Carlo, Elmo, Dina.

For circular arrangements, fix the facing convention before placing anyone. Seated facing the centre, a person's right-hand side runs anticlockwise as seen from above; seated facing outward, it runs clockwise. Getting this backwards inverts the whole answer.

Ranking puzzles

Collapse every clue into one chain of inequalities. Five examinees — Aira, Ben, Cielo, Diego and Elena. Elena scored higher than Diego; Diego scored higher than Aira; Ben scored lower than Aira; Cielo scored lowest. The chain reads Elena > Diego > Aira > Ben > Cielo, so the third-highest scorer is Aira.

Two phrasings trip candidates constantly. 'Only two examinees scored higher than Nena' puts Nena third from the top, not second. And if Nena is third from the bottom of seven, she is fifth from the top, because position from the top equals $7 - 3 + 1$.

Blood-relation puzzles

Draw a small family tree and collapse each phrase from the inside out. The key move is recognising self-references.

Worked example. Rafael, a man, said: 'Teresa's father is the only son of my father.' Work inward. 'The only son of my father' can only be Rafael himself, since Rafael is male and his father has no other son. So Teresa's father is Rafael, which makes Rafael the father of Teresa.

A second pattern: Liway, a woman, introduced a man by saying 'his wife is the only daughter of my mother'. Liway's mother's only daughter is Liway, so the man's wife is Liway and he is Liway's husband.

Common traps

• Locking onto the first shared property you notice instead of counting how many members it covers.
• Accepting a three-against-two split as the answer.
• Forgetting to propagate a confirmed match across both the row and the column, and then into the other sub-grids.
• Reading 'to the right of' as 'immediately to the right of'.
• Assuming a gender in a blood-relation item that the wording never supplies.
• Answering a ranking item by counting from the wrong end of the list.