Ethan's Notes

    Date To Day

    #Math

    Resources

    Mike Boyd Video
    Numberphile Video

    Formula

    $\displaystyle d\equiv f\left( {\left\lfloor \frac{Y}{100}\right\rfloor} \right)+g(\text{mod}(Y,100))+D-D_{D}~\left(\text{mod }7\right)$

    • $\displaystyle d$ is the day in number form
      • Sunday = 0 (Noneday)
      • Monday = 1 (Oneday
      • Tuesday = 2 (Twosday)
      • Wednesday = 3 (Triday)
      • Thursday = 4 (Foursday)
      • Friday = 5 (Fiveday)
      • Saturday = 6 (Sixday)
    • $\displaystyle Y$ is the year
    • $\displaystyle {\left\lfloor \frac{Y}{100}\right\rfloor}$ removes the last two digits of $\displaystyle Y$. For example, $\displaystyle {\left\lfloor \frac{1984}{100}\right\rfloor}$ is 19
    • $$
      f(x)=
      \begin{cases}
      0 & x\equiv 1~\left(\text{mod }4\right) \\ 5 & x\equiv 2~\left(\text{mod }4\right) \\ 3 & x\equiv 3~\left(\text{mod }4\right) \\ 2 & x\equiv 0~\left(\text{mod }4\right)
      \end{cases}
      $$
    • $\displaystyle \text{mod}\left(Y,100\right)$ takes the last two digits of $\displaystyle Y$. For example, $\displaystyle \text{mod}\left(1984,100\right)=84$
    • $\displaystyle g(x)\equiv {\left\lfloor \frac{x}{4}\right\rfloor}+x$
      • Can simplify using Conway's method of subdividing years by 12 (see additional notes)
    • $\displaystyle D$ is the date number
    • $\displaystyle D_{D}$ is the Doomsday number for the particular month
      • For months 1 (January), 2 (February), ..., and etc., $\displaystyle D_{D}$ is as follows:
      1. 3 (4 for leap years)
      2. 28 (29 for leap years)
      3. 14 ($\displaystyle \pi$-day)
      4. 4
      5. 9
      6. 6
      7. 11
      8. 8
      9. 5
      10. 10
      11. 7
      12. 12
    • E.g: What day was August 15th, 1930?
      *$$
      \begin{align}
      d &\equiv f\left( {\left\lfloor \frac{1930}{100}\right\rfloor} \right)+g(\text{mod}\left(Y,100\right))+\cancelto{ \cancelto{ 0 }{ 7 } }{ 15-8 }~\left(\text{mod }7\right) \\ &= \cancelto{ \cancelto{ 5 }{ 12 } }{ 3+9 }+0 \\ &= 5\rightarrow \text{Friday}
      \end{align}
      $$

    Tools

    Additional Notes

    Dooms Days

    These are dates that all share the same day throughout the year

    • 1/3 (1/4 leap year)
    • 2/7 (2/1 leap year)
    • 3/14
    • 4/4
    • 5/9
    • 6/6
    • 7/11
    • 8/8
    • 9/5
    • 10/10
    • 11/7
    • 12/12

    Dooms Days for Years

    Centuries

    Centuries repeat in the following cycle

    • 1700 0
    • 1800 5
    • 1900 3
    • 2000 2

    Common Years

    • 2001 3
    • 2002 4
    • 2003 5
    • 2004 0
    • 2005 1
    • 2006 2
    • 2007 3
    • 2008 5
    • 2009 6
    • 2010 0
    • 2011 1
    • 2012 3
    • 2023 2

    Conway's Method for Years

    • 0 0
    • 12 1
    • 24 2
    • 36 3
    • 48 4
    • 60 5
    • 72 6
    • 84 7
    • 96 8