Calendar

Module 10 — Lesson 0010 · In Depth · 45 min

Every calendar problem is mod 7 arithmetic. Days repeat every 7, so only the remainder — odd days — matters. Count odd days from a reference, mod 7, map to weekday.

1. The Foundation: Odd Days

Odd Days = Total Days mod 7 — remainder after dividing by 7.

Odd days — mod 7 wheel Mon (0) Tue(1) Wed(2) Thu(3) Fri(4) Sat(5) Sun(6) +3 → Thu Total days mod 7 = shift from start day
Only the remainder moves the weekday.

2. Ordinary vs Leap Years

Type Days Odd Days
Ordinary year 365 =52×7+1 1
Leap year 366 =52×7+2 (Feb 29) 2

Identifying a Leap Year

Leap year decision Year = 1900? Ends with 00? yes → /400? no → /4? yes (2000) → leap (2 odd) no (1900) → not leap (1 odd) yes → leap no → ordinary
Century years need ÷400, not just ÷4.

3. Long-Term Odd Day Cycles — Centuries

100 years = 76 ordinary +24 leap → 76×1 +24×2=124 days → 124 mod7=5 odd days.

Period Calculation Odd Days
100 years 76×1+24×2=124 → 124 mod7 5
200 years 5×2=10 → 10 mod7 3
300 years 5×3=15 → 15 mod7 1
400 years 5×4+1 extra leap =21 → 21 mod7 0

400-year rule: any multiple of 400 years (400,800,1200,1600,2000,2400) → 0 odd days. So 1600 years =0, 2000 years =0, etc.

4. Day-to-Day and Month-to-Month Jumps

Month type Days Odd days
31-day (Jan,Mar,May,Jul,Aug,Oct,Dec) 31=28+3 3
30-day (Apr,Jun,Sep,Nov) 30=28+2 2
February 28 or 29 0 (ordinary) / 1 (leap)

"Same Day" rule: In an ordinary year, Jan1 and Dec31 are same weekday; in a leap year, Dec31 is one day ahead of Jan1 (because Feb adds 1 odd day before December).

5. Repetition of Calendars

A year's calendar repeats when cumulative odd days since that year ≡0 mod7.

If year is… Repeats after Example
Leap year 28 years 2024 → 2052
Leap year +1 6 years 2025 → 2031
Leap year +2 11 years 2026 → 2037
Leap year +3 11 years 2027 → 2038

6. Systematic Solving Methodology

  1. Reference day: Jan 1, 0001 AD was Monday. Codes: 0=Sun,1=Mon,2=Tue,3=Wed,4=Thu,5=Fri,6=Sat.
  2. Break down years: groups of 400, then 100, then remaining years. Example: 1947 years =1600+300+46 completed years + target year 1947.
  3. Leap years in remainder: leaps = ⌊remaining/4⌋ − centuries correction → total odd days = (ordinary×1 + leaps×2) mod7 or simply (remaining + leaps) mod7.
  4. Add monthly odd days: Jan 1 to target date within target year.
  5. Final sum mod7: map remainder to weekday via day codes.
Worked: What day was Aug 15, 1947?
Years 0→1946: 1600(0)+300(1)+46 years. Leaps in 46 =11 (46/4). Odd =46+11=57 → 57 mod7=1. Centuries 1 + remainder 1 =2 odd days for years.
Months Jan→Jul 1947 (1947 ordinary): Jan3+Feb0+Mar3+Apr2+May3+Jun2+Jul3=16 → 2 odd days. Days Aug 15 → 15→ 1 odd day (15 mod7=1, but counting from Aug1: 15 days includes Aug15, careful offset). Total 2+2+1=5 → Friday (0=Sun→5=Fri) — indeed India's Independence Day was Friday.

7. Summary of Variables

Symbol Meaning
Y Total years
L Number of leap years in period
O Number of ordinary years
Total Odd = 1×O +2×L All periods mod7
Century codes 100→5, 200→3, 300→1, 400→0

Odd days for total days N is?

Ordinary year has how many odd days?

Leap year has how many odd days?

1900 is leap?

100 years has how many odd days?

400 years has how many odd days?

31-day month has how many odd days?

Leap year calendar repeats after?

Notes

Calendar :- Basics :-

Odd days ⇒ total mod7 ; 0→same weekday

Ordinary 365→1 ; Leap 366→2 (Feb 29)

Leap: non-century /4 ; century /400 (1900×, 2000✓)

Cycles :-

100y→5 ; 200y→3 ; 300y→1 ; 400y→0 (multiples of 400→0)

Months: 31→3, 30→2, Feb 0/1 ; same-day rule: ordinary Jan1=Dec31, leap Dec31+1

Repetition: leap→28, +1→6, +2→11, +3→11

Method :-

Ref Jan1 0001 Mon (0=Sun) ; break years into 400/100/remainder ; leaps=⌊rem/4⌋ ; total=(rem+leaps) mod7 ; add monthly odds → final mod7 → weekday

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