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Mathematical Reasoning: Puzzles You Can Solve with Rules

Reasoning questions test whether you can spot a rule and use it. Odd one out: find the rule most items share. Syllogism: draw circles for "all", "some" and "no" and accept only what MUST be true. Blood relations: draw a family tree. Coding-decoding: find how letters or numbers shift. Cryptarithms: digits hide behind letters; use place value, carries and odd/even (parity) to find them.

🎬 Step-by-step story

  1. Odd one out: look for the rule most items share. 2, 3, 5, 7 and 11 are prime numbers. 9 = 3 Γ— 3 is not, so 9 lifts out.
  2. Syllogism: "All roses are flowers" puts the rose circle inside the flower circle. "Some flowers are red" adds a red circle that only touches the flower circle. Roses and red may not meet, so "some roses are red" is not certain.
  3. Blood relations: draw a family tree. Cubes are male, balls are female. Riya's father's sister is Riya's aunt. Aman, the grandson of Grandpa, is Riya's brother.
  4. Coding: every letter moves 2 places forward in the alphabet. C becomes E, A becomes C, T becomes V. So CAT is written ECV.
  5. Digits in disguise: A + A + A = BA. Three A's end in A, so 2A must end in 0, and A = 5, B = 1. Parity: an odd pile plus an odd pile always pairs up into an even number.
  6. Your turn: pick a word and a shift. Read its code, then shift back to decode it.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

πŸ€” Common doubts, cleared

What if two different rules both give an answer in odd one out?

Choose the rule that is simplest and fits all the other items exactly, like "prime numbers" in the 3D.

Why does "some roses are red" not follow?

In the circle picture, the red circle can sit on the part of flowers where there are no roses. Since one picture breaks it, it is not certain.

Do I need to know the gender to name a relation?

Often yes: a brother's child could be a nephew or a niece. The tree uses cubes and balls so you never forget.

What happens after Z in a shift code?

Go back to the start: Y + 2 = A. The alphabet is a circle. Try shifting BRAIN in the free play.

Why must A be 5 in A + A + A = BA?

The units digit of 3A must be A, so 2A ends in 0. Only A = 5 (A = 0 is not allowed) works.

How do I decode a message?

Use the opposite shift. A +2 code is undone by βˆ’2.

Odd one out and number series

In an odd one out (odd man out) question, all items but one follow the same rule. Check common rules in this order:

Example: 121, 144, 169, 180, 196 β†’ 180 is not a perfect square.

In a series, find the gap: 3, 7, 15, 31, ? β†’ gaps 4, 8, 16, so next gap 32 β†’ 63. Also try Γ—2 + 1.

Syllogism with circles

A syllogism gives statements (taken as true even if they sound strange) and asks which conclusions must follow.

Draw the smallest picture that fits every statement. A conclusion follows only if it is true in every possible picture. If one picture breaks it, it does not follow. When two opposite conclusions ("Some A are C" / "No A is C") cover all cases, the answer is "either one follows".

Blood relations

Draw a tree: one generation per row, a square for male, a circle for female, a horizontal line for husband–wife, a vertical line for parent–child.

Work from the end of the sentence back to the start: "A is the son of the brother of B's mother" β†’ B's mother β†’ her brother (B's uncle) β†’ his son, so A is B's cousin.

Coding and decoding

Each letter can be given a number: A = 1, B = 2 … Z = 26. Opposite letters add to 27 (A↔Z, B↔Y).

Test the rule on every letter before using it.

Digits in disguise and parity

In a cryptarithm each letter stands for one digit; different letters, different digits; a number does not start with 0.

Tools: (1) look at the units column first, (2) watch carries, (3) the first digit of a longer answer is often 1, (4) use parity.

Parity means even or odd. Even + even = even, odd + odd = even, even + odd = odd. Odd Γ— odd = odd; anything Γ— even = even. Example: can five odd numbers add to 20? No: five odds give an odd total.

Worked: AB + BA = 121 β†’ (10A + B) + (10B + A) = 11(A + B) = 121, so A + B = 11 (e.g. 29 + 92).

Try it at home

Write a secret message with a +3 shift and give it to a friend with only one clue. Draw your own family tree up to your grandparents and make three relation riddles from it.

Key formulas and definitions

Worked examples

1. Odd one out: 8, 27, 64, 100, 125.

8 = 2Β³, 27 = 3Β³, 64 = 4Β³, 125 = 5Β³ are cubes. 100 is not a cube. Answer: 100.

2. Statements: All pens are tools. No tool is food. Conclusions: (I) No pen is food. (II) Some tools are pens.

Pen circle inside tool circle; food circle away from tools. So no pen touches food: I follows. Pens are inside tools, so some tools are pens: II follows. Both follow.

3. Pointing to a man, Sara says, "He is the son of my grandfather's only son." How is he related to Sara?

Grandfather's only son = Sara's father. His son = Sara's brother.

4. If MATH is coded as NBUI, how is CODE coded?

Each letter +1: M→N, A→B, T→U, H→I. So CODE → DPEF.

5. Solve AB + BA = 132 for the largest possible AB.

11(A + B) = 132, so A + B = 12. Largest AB: A = 9, B = 3 β†’ 93 (93 + 39 = 132).

6. Can 7 odd numbers add up to 50?

Odd + odd is even, so 6 of them make an even total; adding the 7th odd makes it odd. 50 is even, so it is impossible.

Common mistakes

Practice quiz

1. Odd one out: 4, 9, 16, 20, 25
2. "All A are B" is drawn as:
3. In a +1 code, DOG is:
4. Odd + odd is always:
5. My mother's brother is my:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is mathematical reasoning?

Using clear rules to reach answers that must be true: spotting patterns, drawing diagrams for statements, and testing each step.

How do I solve syllogism questions fast?

Draw circles for each statement and accept a conclusion only if it is true in every possible drawing.

What is a cryptarithm?

A sum where digits are replaced by letters. You use place value, carries and parity to find each digit.

Where this is taught

CBSE (India)Class 8Number Play
CBSE (India)Class 11Algebra

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