### Computing The Day of the Week (DoW) Any Number of Days (±N) in Relative Past or Future From Any Given Known Starting Point

Suppose we know a certain date is a Thursday (DoWa = 4) and we wish to find (DoWb = ?) for a date N days before or after that date.  This algorithm handles that type of problem and applies to both the Julian and Gregorian calendars.

Given any known starting weekday (DoWa), the following algorithm will compute the day of the week (DoWb) any number of days (±N) in the relative past or future from that starting point.

Algorithm 7:

Starting on any weekday (DoWa), compute the weekday (DoWb)  ±N days from that day
.
```
if N < 0 then SignVal = -1 else SignVal = 1

DoWb = (DoWa + 7 + SignVal*(abs(N) mod 7)) mod 7

```
or
```
SignVal = (N < 0)? -1:1

DoWb = (DoWa + 7 + SignVal*(abs(N) mod 7)) mod 7

```
or
```
DoWb = (DoWa + 7 + ((N < 0)? -1:1)*(abs(N) mod 7)) mod 7

```

In PHP, it can be written as
```\$DoWb = (\$DoWa + 7 + ((\$N < 0)? -1:1)*(abs(\$N) % 7)) % 7;
```

DoW Indices:
0=Sun, 1=Mon, 2=Tue, 3=Wed, 4=Thu, 5=Fri and 6=Sat

EXAMPLE 1:
Starting on any Tuesday (DoWa=2), what would be the day of the week (DoWb) exactly 2013 days in the future?
``` DoWa = 2 (Tuesday)

N = 2013
```
The algorithm gives:
```
SignVal = (2013 < 0)? -1:1 = 1

DoWb = (2 + 7 + 1*(abs(2013) mod 7)) mod 7 = 13 mod 7 = 6 (Saturday)

```
So, starting on any Tuesday, exactly 2013 days later it will be a Saturday, even though we may not know the actual calendar date itself.

EXAMPLE 2:
Starting on any Friday (DoWa=5), what would be the day of the week (DoWb) exactly 5717 days in the past?
``` DoWa = 5 (Friday)

N = -5717
```
The algorithm gives:
```
SignVal = -1

DoWb = (5 + 7 + -1*(abs(-5717) mod 7)) mod 7 = 7 mod 7 = 0 (Sunday)

```

So, starting on any Friday, exactly 5717 days earlier would be a Sunday.

© PHP Science Labs - 2017