Logical and Analytical Reasoning

Analytical reasoning questions test careful, systematic thinking rather than knowledge. The common types each have a reliable method.

Syllogisms

Draw Venn diagrams for the premises and accept a conclusion only if it is true in every possible diagram.

Example: "All doctors are graduates. Some graduates are artists."

• "Some doctors are artists" does not follow: the artists could all be non-doctor graduates.
• "Some graduates are doctors" does follow (as long as doctors exist), because the doctor circle lies inside the graduate circle.

When two conclusions are "Some A are C" and "No A is C" and neither follows on its own, exactly one of them must be true: the answer is "either one or the other follows".

Direction sense

Track the position with coordinates. Facing north, "turn right" means east, and so on. A person walks 5 km north, turns right and walks 3 km, turns right and walks 9 km, then turns left and walks 2 km.

North +5, east +3, south −9, east +2: the final position is (5, −4), so the person is √(5² + 4²) = √41 ≈ 6.40 km away, towards the south-east of the start.

Blood relations

Decode the sentence from the inside out. "Pointing to a man, Riya says: his mother is the only daughter of my mother." The only daughter of Riya's mother is Riya herself, so Riya is the man's mother. Drawing a small family tree (with generations as levels) avoids most mistakes:

Seating arrangements

List the positions, place the fixed clues first, and test the remaining cases one by one. Five friends A, B, C, D and E sit in a row facing north (positions 1 to 5 from the left):

1. C sits in the middle.
2. A does not sit at either end.
3. B sits immediately to the right of D.
4. E does not sit next to C.
5. D sits somewhere to the left of A.

C is at 3, and E (not next to C) must be at 1 or 5. The pair D B needs two adjacent seats, which leaves only seats 1–2 (with E at 5 and A at 4) or 4–5 (with E at 1 and A at 2). Clue 5 rules out the second case. So the order is D B C A E. Checking every arrangement by program confirms the answer is unique:

from itertools import permutations

solutions = []
for order in permutations("ABCDE"):
    pos = {person: i + 1 for i, person in enumerate(order)}
    if (pos["C"] == 3 and pos["A"] not in (1, 5) and pos["B"] == pos["D"] + 1
            and abs(pos["E"] - pos["C"]) != 1 and pos["D"] < pos["A"]):
        solutions.append(" ".join(order))
print(solutions)

['D B C A E']

Without clue 5 there would be two answers (D B C A E and E A C D B), so a well-posed puzzle needs every clue.

Coding and decoding

Find the rule from the example, then apply it. If COMPUTER is written EQORWVGT, each letter has moved 2 places forward in the alphabet (C → E, O → Q, …), so GATE becomes ICVG. Other common rules: reversing the word, moving letters by their position number, or replacing letters by their alphabet positions (A = 1, …, Z = 26).

Number and letter series

Look at the differences, then at the differences of the differences, ratios, squares and cubes, or alternate terms:

• 2, 6, 12, 20, 30, 42: differences 4, 6, 8, 10, 12 (the terms are n(n + 1)).
• 3, 7, 15, 31, 63: each term is 2 × previous + 1 (the terms are 2ⁿ − 1).
• 1, 4, 9, 16, 25, 36: squares.
• 2, 3, 5, 7, 11, 13: primes.

Practice questions

from itertools import permutations
sols = []
for p in permutations("PQRSTU"):
    pos = {c: i + 1 for i, c in enumerate(p)}
    if (pos["P"] in (1, 6) and pos["T"] == 4 and pos["S"] == pos["T"] - 2 and pos["Q"] == pos["R"] + 1
            and pos["U"] not in (1, 6) and pos["R"] > pos["U"]):
        sols.append(p)
print(sols[0][5] if len(sols) == 1 else "not unique")

print("".join(chr((ord(c) - 65 - 1) % 26 + 65) for c in "LOGIC"))