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Century Code & Day Formula (Zeller’s Rule)

Introduction

Century codes और day formulas (जैसे Zeller’s Rule) आपको किसी भी Gregorian कैलेंडर तिथि के लिए weekday तेज़ी से और विश्वसनीय रूप से निकालने देते हैं। यह पैटर्न महत्वपूर्ण है क्योंकि यह एक formula-आधारित, परीक्षा-मैत्री तरीका देता है जो वर्ष-दर-वर्ष गिनती से बचाता है और शताब्दी-सीमाओं के पार तारीखों पर भी अच्छी तरह काम करता है।

Pattern: Century Code & Day Formula (Zeller’s Rule)

Pattern

मुख्य विचार: तारीख को घटकों में बदलें (दिन, समायोजित महीना, century का year-of-century और century), month और century codes तथा वर्ष के योगदान और leap समायोजन लागू करें, सबको जोड़ें और modulo 7 लें ताकि weekday मिल सके।

हम Zeller’s congruence (Gregorian कैलेंडर) का उपयोग करते हैं जहाँ महीने को इस तरह लिया जाता है: March = 3 … December = 12, और January/February को पिछले वर्ष के महीने 13/14 माना जाता है।
इस सूत्र का प्रयोग करें: h = ( q + floor(13*(m+1)/5) + K + floor(K/4) + floor(J/4) + 5*J ) mod 7
जहाँ:

  • q = महीने का दिन
  • m = महीना (3 = Mar, …, 12 = Dec; Jan = 13, Feb = 14 पिछले वर्ष के)
  • K = year of century (year % 100)
  • J = zero-based century (floor(year / 100))
  • h = 0 → Saturday, 1 → Sunday, 2 → Monday, … 6 → Friday

Step-by-Step Example

Question

26 January 1950 किस दिन था?

Solution

  1. Step 1: Zeller के लिए महीना और वर्ष समायोजित करें

    January को पिछले वर्ष का महीना 13 मानें। तो उपयोग करें: q = 26, m = 13, और year = 1949।

  2. Step 2: K और J निकालें

    K = year % 100 = 1949 % 100 = 49.
    J = floor(year / 100) = floor(1949 / 100) = 19.

  3. Step 3: सूत्र के पुर्जे निकालें

    • floor(13*(m+1)/5) = floor(13*(14)/5) = floor(182/5) = 36
    • floor(K/4) = floor(49/4) = 12
    • floor(J/4) = floor(19/4) = 4
    • 5*J = 5*19 = 95
    अब पदों का योग लें: h = ( q + 36 + K + 12 + 4 + 95 ) mod 7 = (26 + 36 + 49 + 12 + 4 + 95) mod 7.
    कुल = 222.

  4. Step 4: modulo 7 लें

    222 mod 7 = 5. Zeller के अनुसार h = 5 का मतलब है Thursday (0 = Sat, 1 = Sun, 2 = Mon, 3 = Tue, 4 = Wed, 5 = Thu, 6 = Fri).

  5. Final Answer:

    26 January 1950 Thursday था।

  6. Quick Check:

    alternate odd-day विधि या ऐतिहासिक कैलेंडर से cross-check करें - दोनों पुष्टि करते हैं कि यह Thursday था। ✅

Quick Variations

1. Zeller’s में महीनों को Mar→3 … Jan→13, Feb→14 मानें (और Jan/Feb के लिए वर्ष घटाएँ)।

2. तेज़ मानसिक गणना के लिए आप month codes और century codes पहले से याद कर सकते हैं, और फिर सरल रूप का उपयोग करें: weekday = (day + month_code + year_code + century_code + leap_adjust) mod 7।

3. Tomohiko Sakamoto का algorithm या Doomsday rule भी वैकल्पिक तरीके हैं जो मानसिक गणना में तेज़ हो सकते हैं; जो तरीका आपको बेहतर सूट करे वह चुनें।

4. हमेशा अपने फ़ॉर्मूले में उपयोग होने वाले weekday mapping को सत्यापित करें (Zeller में Saturday = 0 होता है)।

Trick to Always Use

  • अगर month ≤ 2 (जैसे Jan/Feb) हो तो month में +12 करें और year से 1 घटा कर K तथा J निकालें।
  • floor(13*(m+1)/5) के मान (month offsets) को m = 3..14 के लिए पहले से निकाल कर रखें ताकि बार-बार भाग न करना पड़े।
  • आम सदीयों के लिए 5*J और floor(J/4) के adjustments याद रखें ताकि गणना तेज़ हो।
  • बड़े मध्यवर्ती योगों को जल्दी ही mod 7 कर दें ताकि अंकगणित छोटा रहे और त्रुटि कम हो।

Summary

Summary

  • तिथि को q, समायोजित m, K, और J में बदलें। (Jan/Feb के लिए महीने 13/14 और पिछले वर्ष का प्रयोग करें.)
  • Zeller का सूत्र लागू करें: h = ( q + floor(13*(m+1)/5) + K + floor(K/4) + floor(J/4) + 5*J ) mod 7.
  • h को weekday के रूप में पढ़ें जहाँ 0 = Sat, 1 = Sun, …, 6 = Fri.
  • मासिक/सदी कोड पहले से रख लें और अक्सर mod 7 करके मैन्युअल काम को सरल बनाएं।

याद रखने के लिए उदाहरण: 26 January 1950 → Thursday.

Practice

(1/5)
1. What day of the week was 1st January 2000?
easy
A. Saturday
B. Sunday
C. Friday
D. Monday

Solution

  1. Step 1: Adjust month/year for Zeller

    January → treat as month 13 of previous year. So use: q = 1, m = 13, year = 1999.

  2. Step 2: Compute K and J

    K = 1999 % 100 = 99, J = floor(1999 / 100) = 19.

  3. Step 3: Evaluate formula components

    floor(13*(m+1)/5) = floor(13*14/5) = 36;
    floor(K/4)=floor(99/4)=24;
    floor(J/4)=floor(19/4)=4;
    5*J = 95.
    Sum: h = (1 + 36 + 99 + 24 + 4 + 95) = 259 → 259 mod 7 = 0.

  4. Step 4: Map result to weekday

    Zeller mapping: 0 → Saturday. So 1 Jan 2000 = Saturday.

  5. Final Answer:

    Saturday → Option A

  6. Quick Check:

    Known anchor: 1 Jan 2000 (Y2K) was Saturday ✅
Hint: For Jan/Feb treat them as months 13/14 of previous year; apply Zeller and map 0→Sat.
Common Mistakes: Forgetting to decrement year for Jan/Feb or mis-mapping Zeller's h value.
2. What day of the week was 15th August 1947?
easy
A. Thursday
B. Friday
C. Saturday
D. Sunday

Solution

  1. Step 1: Use Zeller (month > 2 so no year change)

    q = 15, m = 8, year = 1947.

  2. Step 2: Compute K and J

    K = 1947 % 100 = 47, J = floor(1947/100) = 19.

  3. Step 3: Evaluate components

    floor(13*(m+1)/5) = floor(13*9/5) = 23;
    floor(K/4)=floor(47/4)=11;
    floor(J/4)=4;
    5J=95.
    Sum: h = (15 + 23 + 47 + 11 + 4 + 95) = 195 → 195 mod 7 = 6.

  4. Step 4: Map to weekday

    Zeller mapping: 6 → Friday. So 15 Aug 1947 = Friday.

  5. Final Answer:

    Friday → Option B

  6. Quick Check:

    Historical calendars confirm 15-Aug-1947 was Friday ✅
Hint: Cross-check Zeller with an odd-day count or known historical anchor if unsure.
Common Mistakes: Indexing errors with month codes or misreading Zeller's mapping.
3. What day of the week was 1st January 2010?
easy
A. Thursday
B. Saturday
C. Friday
D. Sunday

Solution

  1. Step 1: Adjust for January

    Use q = 1, m = 13, year = 2009 (Jan treated as month 13 of previous year).

  2. Step 2: Compute K and J

    K = 2009 % 100 = 9, J = floor(2009/100) = 20.

  3. Step 3: Evaluate components

    floor(13*(m+1)/5) = 36;
    floor(K/4)=floor(9/4)=2;
    floor(J/4)=floor(20/4)=5;
    5J=100.
    Sum: h = (1 + 36 + 9 + 2 + 5 + 100) = 153 → 153 mod 7 = 6.

  4. Step 4: Map to weekday

    Zeller mapping: 6 → Friday. So 1 Jan 2010 = Friday.

  5. Final Answer:

    Friday → Option C

  6. Quick Check:

    New Year 2010 is known to be Friday ✅
Hint: Remember to decrement the year for Jan/Feb conversions before computing K and J.
Common Mistakes: Forgetting the Jan/Feb adjustment or misinterpreting h→weekday mapping.
4. What day of the week was 29th February 2000 (leap day)?
medium
A. Monday
B. Wednesday
C. Thursday
D. Tuesday

Solution

  1. Step 1: Treat February as month 14 of previous year

    For Zeller use q = 29, m = 14, year = 1999.

  2. Step 2: Compute K and J

    K = 1999 % 100 = 99, J = floor(1999/100) = 19.

  3. Step 3: Evaluate components

    floor(13*(m+1)/5) = floor(13*15/5) = 39;
    floor(K/4)=24;
    floor(J/4)=4;
    5J=95.
    Sum: h = (29 + 39 + 99 + 24 + 4 + 95) = 290 → 290 mod 7 = 3.

  4. Step 4: Map to weekday

    Zeller mapping: 3 → Tuesday. So 29 Feb 2000 = Tuesday.

  5. Final Answer:

    Tuesday → Option D

  6. Quick Check:

    Historical references list 29-Feb-2000 as Tuesday ✅
Hint: For Feb in leap years use month = 14 and previous year; reduce modulo 7 early to simplify arithmetic.
Common Mistakes: Mismapping Zeller's h to weekday names or forgetting to shift year for Jan/Feb.
5. What day of the week was 31st December 1999?
medium
A. Friday
B. Thursday
C. Saturday
D. Sunday

Solution

  1. Step 1: Month > 2 (no adjustment)

    q = 31, m = 12, year = 1999.

  2. Step 2: Compute K and J

    K = 99, J = 19.

  3. Step 3: Evaluate formula pieces

    floor(13*(12+1)/5)=33;
    floor(K/4)=24;
    floor(J/4)=4;
    5J=95.
    Sum: h = (31 + 33 + 99 + 24 + 4 + 95) = 286 → 286 mod 7 = 6.

  4. Step 4: Map to weekday

    Zeller mapping: 6 → Friday. So 31 Dec 1999 = Friday.

  5. Final Answer:

    Friday → Option A

  6. Quick Check:

    Day before 1 Jan 2000 (Saturday) is Friday ✅
Hint: After computing h, map Zeller's index carefully (0=Sat,…,6=Fri) and cross-check adjacent known dates.
Common Mistakes: Reading h directly as conventional weekday without applying the mapping.

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