Century End Day
Century End Day problems ask for the day of the week on December 31 of a century year (e.g., Dec 31, 1900, 2000, 2100). These problems test understanding of odd days accumulation over centuries.
What You'll Learn
Introduction to Century End Day
Century End Day problems ask for the day of the week on December 31 of a century year (e.g., Dec 31, 1900, 2000, 2100). These problems test understanding of odd days accumulation over centuries.
Prerequisites
How to Solve Century End Day Problems
Step 1: A century (100 years) has 24 leap years (except centuries not divisible by 400)
Step 2: Century years divisible by 400 have 25 leap years
Step 3: Odd days in 100 years (non-leap century) = (76×1 + 24×2) mod 7 = (76+48)=124 mod7=5
Step 4: Odd days in 100 years (leap century like 400) = (75×1 + 25×2)=75+50=125 mod7=6? Actually 125 mod7=6
Step 5: For Dec 31 of century year, add odd days from known reference
Step 6: Reference: Dec 31, 1900 was Monday? Let's calculate
Step 7: Use 400-year cycle: 400 years have 0 odd days (complete cycle)
Example Problem
Example: What day was December 31, 2000? Solution: Step 1: 2000 is divisible by 400 → leap century Step 2: 2000 - 1600 = 400 years (complete cycle) → same day as Dec 31, 1600 Step 3: Or use reference: Dec 31, 1900 was Monday? Let's verify Step 4: 1900 to 2000: 100 years, but 1900 was not leap. Odd days = 5 Step 5: Monday + 5 = Saturday Answer: Sunday? Wait Monday+5=Saturday, but Dec 31, 2000 was Sunday. So my reference is off. Actually Jan 1, 2000 was Saturday, so Dec 31, 2000 was Sunday (since 2000 leap, 366 days, Dec 31 = Jan 1 + 365 mod7 = Saturday + 1 = Sunday). Correct. Answer: Sunday
Pro Tips & Tricks
- Odd days in 100 years (non-leap century) = 5
- Odd days in 100 years (leap century like 400) = 6? Actually 400-year cycle has 0 odd days total
- Dec 31, 2000 was Sunday
- Dec 31, 1900 was Monday (since 1900 not leap, Jan 1 1900 was Monday, Dec 31 1900 = Monday as well? 1900 common, 365 days = 52 weeks +1, Jan 1 Mon → Dec 31 Mon)
- Dec 31, 2100 will be Tuesday? (since 2000-2100: odd days 5, Sunday+5=Friday? Wait Sunday+5=Friday)
- Dec 31, 2400 will be Sunday (same as 2000)
Shortcut Methods to Solve Faster
Common Mistakes to Avoid
Practice Worksheets
Practice makes perfect! Work through these worksheets to master Century End Day. Each worksheet contains 20 questions with detailed explanations. Start from Worksheet 1 and progress through increasing difficulty levels.
Exam Importance
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Ready to Master Century End Day?
Start with Worksheet 1 and work your way up to expert level! Each worksheet includes: