Day-of-Week Calculation Methods

Introduction

Both methods allow you to calculate the day of the week for any date using mental math. The core principle is the same: break a date into components (Century, Year, Month, Day), look up or calculate a code for each component, add all codes together, and take Mod 7 to get the day of the week.

The key difference is how the YEAR component is handled: the 10s Method uses Base-10 (decades) while the 20s Method uses Base-20. Both produce the Year Code by adding an Anchor Code and a Remainder Code. Once the Year Code is determined, the rest is a cakewalk — calculating the Year Code mentally is the hardest part.

Method 1: Zakariya Latif 10s Method (Base-10 / Decade System)

The 10s Method divides the 2-digit year into two parts: an Anchor Year (nearest lower multiple of 10) and a Unit Year (0–9). This method has a special Latif Patch Rule that adjusts codes for odd decades. Adding the Anchor Code (Table 1.1) and the Unit Code (Table 1.2) gives the Year Code, which is then combined with the Century, Month, and Day codes from the Shared Tables.

Steps for the 10s Method

Table 1.1 — 10s Anchor Year Codes

💡 Pattern: subtract 2 and 1 alternately from the previous code. Apply MOD 7 when the result is negative.

Table 1.2 — Unit Year Codes (with Latif Patch applied)

💡 Unit Year Code formula: (Unit Year + INT(Unit Year / 4)) MOD 7

Method 2: Zakariya Latif 20s Method (Base-20 System)

The 20s Method divides the 2-digit year into an Anchor Year (nearest lower multiple of 20) and a Remainder (0–19). Adding the Anchor Code and the Remainder Code gives the Year Code, which combined with the Century, Month, and Day codes gives the day of the week.

This method is simpler — no patch rule needed! The remainder formula handles leap years automatically.

Steps for the 20s Method

Table 2.1 — 20s Anchor Year Codes

💡 Pattern: simply subtract 3 from the previous code and apply MOD 7. Only 5 codes to memorize!

Table 2.2 — Remainder-Year Codes (0–19)

💡 The INT(R/4) automatically counts leap years, adding +1 for every 4 years.

Shared Tables: Century, Month & Day Codes

Table 3 — Century Codes

💡 Codes cycle through 0, 5, 3, 1 repeatedly. Centuries 16, 20, 24, 28 = 0; 17, 21, 25, 29 = 5; and so on.
Formula: (5 × (Century mod 4)) mod 7

Table 4 — Month Codes

💡 For leap years, subtract 1 from Jan & Feb codes only.
Formula: Raw Code = (Days before month) MOD 7. The Final Code includes the −1 adjustment.

Table 5 — Day Codes (final result mapping)

💡 The final formula gives a number 0–6; match it here to get the day. Day-of-month code example: 15th of any month ⇒ 15 mod 7 ⇒ 1.

Worked Example: What day was January 15, 1995?

Using the 10s Method

Using the 20s Method

✅ Both methods give the same answer! 💡 Keep doing Mod 7 whenever the day of the month or any sum grows larger than 7.

Quick Reference Cheat Sheet

Important Notes