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# How to Figure the Day of the Week When a Date Occurs

For any given date in the Gregorian calendar system (the "leap year" system that we currently use), it is possible to determine the day of the week on which that date occurs. Even with the existence of leap years and the fact that some months have more days than others, you can apply a mathematical formula to determine whether a date occurred on a Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, or Saturday.

To use the formula, all you need is the month, day, and 4-digit year number. You can also use the convenient calculator on the left to find the day of the week.

#### Step 1

Call the year number**Y**, the month number

**M**, and the day of the month

**D**. In the cases of January and February, you should consider January to be the 13th month of the previous year and February the 14th month of the previous year. So for January you set

**M**= 13 and subtract 1 from

**Y**. For February you set

**M**= 14 and also subtract 1 from

**Y**.

This adjustment places February at the end of the yearly cycle. For purposes of mathematical calculations, it is more convenient to consider February the last month since it has a variable number of days.

(When entering dates in the calculator, do not worry about subtracting 1 from the year. The calculator takes care of these adjustments for you. If you want to find the day of the week for February 1, 1999, enter 1999 as the year, not 1998.)

#### Step 2

Compute the following sum:D + 2M + ⌊(3M+3)/5⌋ + Y + ⌊Y/4⌋ - ⌊Y/100⌋ + ⌊Y/400⌋

For example, if you choose the date February 1, 1999, then you compute

1 + 2(14) + ⌊(3(14)+3)/5⌋ + 1998 + ⌊1998/4⌋ - ⌊1998/100⌋ + ⌊1998/400⌋

= 1 + 28 + 9 + 1998 + 499 - 19 + 4

= 2520

#### Step 3

Take the number you computed in Step 2 and divide by 7. Then look at the remainder**R**. The value of the remainder gives you the day of the week on which the date occurred.

R = 0: Monday

R = 1: Tuesday

R = 2: Wednesday

R = 3: Thursday

R = 4: Friday

R = 5: Saturday

R = 6: Sunday

Using the example above, 2520/7 has a remainder of 0, so this date falls on a Monday.

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