Introduction
Day-of-Birth puzzles ask you to determine the weekday on which a person was born using information about ages, birthdays, leap years, and weekday relationships. These problems are common in competitive exams because they combine odd-day arithmetic with reasoning about inclusive/exclusive counting and leap-year adjustments.
Pattern: Day-of-Birth Puzzle
Pattern
Key idea: Convert age differences and given weekday facts into total odd days (mod 7), account for leap years in the interval, and shift the known weekday backward or forward to find the birth weekday.
Step-by-Step Example
Question
A person born on 10 January 1990 celebrated his 30th birthday on a Wednesday. On which day of the week was he born?
Solution
Step 1: Translate the age gap into years crossed
His 30th birthday occurs 30 years after birth year span: from 10 Jan 1990 → 10 Jan 2020 (30 years).Step 2: Count leap years in the interval
Leap years between 1990 and 2019 inclusive: 1992, 1996, 2000, 2004, 2008, 2012, 2016 → total = 7 leap years.Step 3: Compute total odd days
Ordinary years = 30 - 7 = 23 → contribute 23 × 1 = 23 odd days. Leap years contribute 7 × 2 = 14 odd days. Total = 23 + 14 = 37 odd days. Reduce mod 7: 37 ÷ 7 → remainder = 2 odd days.Step 4: Determine direction of shift
The 30th birthday weekday is Wednesday. This is 30 years forward from birth, so birth weekday = Wednesday - (total odd days) = Wednesday - 2 days = Monday.Final Answer:
MondayQuick Check:
Birth weekday Monday → +2 odd days over 30 years → Monday + 2 = Wednesday ✅
Quick Variations
1. Age given in years and months - convert full years first, then handle remaining months/days with month-wise odd days.
2. Birthday celebrated on a weekday before the birth date in the calendar year - adjust direction and inclusivity accordingly.
3. When intervals span century boundaries, remember century leap exceptions (only divisible by 400 count as leap).
4. If told the weekday on a particular birthday and asked for weekday on a different birthday, compute odd days between those birthdays and shift accordingly.
Trick to Always Use
- Step 1 → Convert the given birthday difference into a span of whole years (and remaining months/days if any).
- Step 2 → Count leap years inside that span precisely (apply century rule where needed).
- Step 3 → Compute total odd days = ordinary×1 + leap×2, reduce mod 7.
- Step 4 → Shift the known weekday backward (if going from birthday to birth) or forward (if going from birth to birthday) by the odd-day remainder.
Summary
Summary
- Always convert age differences into exact year spans and count leap years precisely.
- Use odd-day arithmetic (ordinary = 1, leap = 2) and reduce modulo 7.
- Carefully choose direction (forward/backward) when applying the shift to the known weekday.
- Quick check by re-applying the odd-day shift to verify the given birthday weekday.
Example to remember:
Born on Monday → 30th birthday on Wednesday (30-year span → +2 odd days)
Hint: Count leap years in the completed years, compute odd days, then shift backward from the known birthday.
Common Mistakes: Including the birthday year (2020) when counting completed leap years.