Analogy

An analogy is a pair of items in some relation, $A : B$, together with a third item $C$; the task is to find the item $D$ that stands in the same relation to $C$: $A : B :: C : D$. The items can be words, numbers, letter groups or figures. The relation required is the one in which $B$ stands to $A$, not merely a word associated with $C$.

The reliable method is to state the relation of the first pair as a short, specific sentence (a "bridge"), then test each option against that sentence. For numbers and letters the bridge is a rule (square, add 1 to each letter's position, reverse the alphabet), which must work for the given pair and produce exactly one option.

Bridge convention. A bridge must be specific enough that only one option fits. "A doctor is related to a hospital" is too weak; "a doctor works in a hospital" is usable.

Direction matters. Petal : Flower (part to whole) is not the same relation as Flower : Petal (whole to part). Keep the order of the given pair in the answer.

Example 1 (worker and workplace). Doctor : Hospital :: Teacher : ?. Bridge: a doctor works in a hospital. A teacher works in a school. Answer: School. "Student" is associated with teacher but fails the bridge.

Example 2 (tool and function). Pen : Write :: Knife : ?. Bridge: a pen is used to write. A knife is used to cut. Answer: Cut. "Sharp" describes a knife but is not its function, and it is an adjective, not a verb.

Example 3 (degree). Drizzle : Downpour :: Breeze : ?. Bridge: a downpour is a much stronger drizzle. A gale is a much stronger breeze. Answer: Gale. "Wind" is a category, not a stronger form.

Example 4 (member and group). Tree : Forest :: Soldier : ?. Bridge: a tree is one member of a forest. Answer: Army. "Uniform" is worn by a soldier; "General" is another member, not the group.

Example 5 (reversed order trap). Flower : Petal :: Book : ?. Bridge: a petal is a part of a flower (whole first). Answer: Page. An option "Library" would reverse the direction (a book is part of a library).

Example 6 (numbers: cube plus one). $3 : 28 :: 4 : 65 :: 5 : ?$. Rule: $n^3 + 1$ ($27 + 1$, $64 + 1$). Answer: $125 + 1 = 126$.

Example 7 (numbers: square minus one). Given $8:63$ and $7:48$, complete $9:?$ using the same familiar rule. The rule $n^2-1$ fits both given pairs and gives 80. The competing rule $8n-1$ fits $8:63$, but gives 55 rather than 48 for 7, so the second pair rejects it. As with any finite analogy, this identifies the intended rule among the candidate rules, not a unique function among all mathematical functions.

Example 8 (letters: shift). ACE : BDF :: GIK : ?. Each letter moves forward one place. Answer: HJL.

Example 9 (letters: opposite letters). CAT : XZG :: DOG : ?. Each letter is replaced by its opposite in the alphabet (positions summing to 27: C = 3 and X = 24). D = 4 becomes 23 = W, O = 15 becomes 12 = L, G = 7 becomes 20 = T. Answer: WLT.

Example 10 (coding). If PAPER is written as QBQFS, how is BOOK written? Each letter moves forward one place. Answer: CPPL.

Example 11 (figures). Figure $A$ is an arrow pointing up with a dot at its tail; figure $B$ is the same arrow pointing right with the dot at its tail. The change is a $90^\circ$ clockwise rotation. Figure $C$ is a triangle with one corner pointing up and a dot inside that corner; $D$ is that triangle turned $90^\circ$ clockwise, with the corner and its dot now pointing right. An option that shows the triangle reflected (corner pointing left) applies the wrong transformation.

Example 12 (double blank). Feather : ? :: Scale : ?. Options: (a) Bird : Fish, (b) Wing : Fin, (c) Light : Heavy, (d) Fly : Swim. Bridge to test: "a feather is the covering of a bird; a scale is the covering of a fish". Only (a) fits both blanks with the same relation. (b) pairs body parts that are not coverings; (c) reads "scale" as a weighing scale; (d) gives actions. With two blanks, write the bridge with both unknowns and test each option pair as a whole.

Example 13 (the pair that breaks the relation). Which pair does not belong: $3 : 27$, $4 : 64$, $5 : 125$, $6 : 206$? The first three are $n : n^3$. For 6 the cube is 216, so $6 : 206$ breaks the relation. The method is the same as for an analogy: find the rule from the majority, then test each pair.

Example 14 (operations on digits). $47 : 28 :: 63 : ?$. Rule: product of the digits ($4 \times 7 = 28$). Answer: $6 \times 3 = 18$. The rule "subtract 19" also turns 47 into 28 but gives 44 from 63; when one rule uses the structure of the number (its digits) and the other is an arbitrary constant, prefer the structural rule, and check the options for the alternative.

Example 15 (opposite-letter clusters). ABC : ZYX :: DEF : ?. Each letter is replaced by its opposite (positions summing to 27: A and Z, B and Y, C and X). D, E, F become W, V, U. Answer: WVU. A constant shift does not fit (A to Z is 1 step back with wrap, B to Y is 3 steps back), so the rule is the opposite-letter map; test a candidate rule on every letter, not just the first.

Example 16 (letters to numbers). CAB : 312 :: BAD : ?. Each letter is written as its position: C = 3, A = 1, B = 2. So BAD is 2, 1, 4: Answer: 214.

Example 17 (abstract nouns). Ignorance : Knowledge :: Poverty : ?. Bridge: ignorance is the absence of knowledge. Poverty is the absence of wealth. Answer: Wealth. "Money" is concrete and is a means, not the state; "Hunger" is an effect of poverty.

Example 18 (figure analogy by counting). Figure $A$ is a triangle containing 1 dot; figure $B$ is a square containing 2 dots. Figure $C$ is a pentagon containing 3 dots. The change from $A$ to $B$ adds one side and one dot, so $D$ is a hexagon containing 4 dots. An option with a hexagon and 3 dots applies only half the change.