Chapter 2: Mod7 Magic

Presidential Calendar Mnemonic (PDF coming soon)
Chapter 1: A Simpler Method
Chapter 2: Mod7 Magic
Chapter 3: The Recipe
Chapter 4: Ten Months
Chapter 5: Zanuary and Zebruary
Chapter 6: US Presidential Elections
Chapter 7: Silly Stories
Chapter 8: Mastery

This is the second post in a series describing the Presidential Calendar Mnemonic. Our goal is to calculate the day of the week, given any date in any century.1

The next step is learning an essential tool called mod7. I prefer the term “mod7 magic” over “mod7 math” because math intimidates some people and mod7 is a bit of a “presto change-o” act. That is, mod7 replaces large numbers with small numbers that achieve the same results.

Let’s use brackets that look kind of like 7s, like this⎡x⎤, to indicate a mod7 operation.2 Think of these 7-brackets as operating in mod7 world, where every number becomes 0 through 6. First, let’s talk about mod7 “floors.” The floors in mod7 world are 0, 7, 14, 21, 28, and 35, obviously multiples of 7. Instead of describing the mod7 function mathematically, let’s simply enumerate all of the answers, so we can see the patterns that emerge.

The Floor 0 group:
⎡ 0 ⎤ = 0 - 0 = 0
⎡ 1 ⎤ = 1 - 0 = 1
⎡ 2 ⎤ = 2 - 0 = 2
⎡ 3 ⎤ = 3 - 0 = 3
⎡ 4 ⎤ = 4 - 0 = 4
⎡ 5 ⎤ = 5 - 0 = 5
⎡ 6 ⎤ = 6 - 0 = 6 = -1 (see the discussion below)

The Floor 7 group:
⎡ 7 ⎤ = 7 - 7 = 0
⎡ 8 ⎤ = 8 - 7 = 1
⎡ 9 ⎤ = 9 - 7 = 2
⎡ 10 ⎤ = 10 - 7 = 3
⎡ 11 ⎤ = 11 - 7 = 4
⎡ 12 ⎤ = 12 - 7 = 5
⎡ 13 ⎤ = 13 - 7 = 6 = -1

The Floor 14 group:
⎡14 ⎤ = 14 - 14 = 0
⎡15 ⎤ = 15 - 14 = 1
⎡16 ⎤ = 16 - 14 = 2
⎡17 ⎤ = 17 - 14 = 3
⎡18 ⎤ = 18 - 14 = 4
⎡19 ⎤ = 19 - 14 = 5
⎡20 ⎤ = 20 - 14 = 6 = -1

The Floor 21 group:
⎡ 21 ⎤ = 21 - 21 = 0
⎡ 22 ⎤ = 22 - 21 = 1
⎡ 23 ⎤ = 23 - 21= 2
⎡ 24 ⎤ = 24 - 21 = 3
⎡ 25 ⎤ = 25 - 21 = 4
⎡ 26 ⎤ = 26 - 21 = 5
⎡ 27 ⎤ = 27 - 21 = 6 = -1

The Floor 28 group:
⎡ 28 ⎤ = 28 - 28 = 0
⎡ 29 ⎤ = 29 - 28 = 1
⎡ 30 ⎤ = 30 - 28 = 2
⎡ 31 ⎤ = 31 - 28 = 3
⎡ 32 ⎤ = 32 - 28 = 4
⎡ 33 ⎤ = 33 - 28 = 5
⎡ 34 ⎤ = 34 - 28 = 6 = -1

The Floor 35 group:
⎡ 35 ⎤ = 35 - 35 = 0
⎡ 36 ⎤ = 36 - 35 = 1
⎡ 37 ⎤ = 37 - 35 = 2
⎡ 38 ⎤ = 38 - 35 = 3
⎡ 39 ⎤ = 39 - 35 = 4
⎡ 40 ⎤ = 40 - 35 = 5
⎡ 41 ⎤ = 41 - 35 = 6 = -1

It’s simply a matter of knowing what group is appropriate, and subtracting the floor. Let’s practice. Answers to the following are in the footnotes.3

⎡ 1 ⎤, ⎡ 5 ⎤, ⎡ 11 ⎤, ⎡ 24 ⎤, ⎡ 17 ⎤, ⎡ 22 ⎤, ⎡ 10 ⎤, ⎡ 20 ⎤, ⎡ 30 ⎤

Now, let’s consider an ideal calendar month. Look on your current year calendar for a month that begins on a Monday, like September of 2025. The 1st is Monday, the 2nd is Tuesday, 3rd is Wednesday, … and the 7th is Sunday. Let’s call this type of calendar a monthType 0. There are six other types of calendar months, based on what day the month starts, and we’ll get to those soon.

Now, let’s combine what we learned in Chapter 1 with a monthType 0, like September 2025.

• What day is the 2nd? Tuesday

• What day is the 9th? ⎡ 9 ⎤ = 9 - 7 = 2. Tuesday

• What day is the 16th? ⎡ 16 ⎤ = 16 - 14 = 2. Tuesday

• What day is the 30th? ⎡ 30 ⎤ = 30 - 28 = 2. Tuesday

This illustrates how 2, 9, 16, and 30 comprise the Tuesday column of a monthType 0 month.

More practice? See the footnotes4 for answers to the following: What days are the 1st, 5th, 11th, 24th, 17th, 22nd, 28th, 10th, 20th, and 30th?

Make your own list of random numbers from 0 to 41, and challenge yourself to do these day code conversions quickly. Again, more speed is more impressive.

There is a bit more mod7 magic to explain. In the table above, any time there is a 6, it can be replaced with -1. In mod7 world, the numbers 6 and -1 are equivalent! Let’s explain by example.

• ⎡ 4 + 0 + 6 ⎤ = ⎡ 10 ⎤ = 10 - 7 = 3. Wednesday

It’s a bit easier if we replace the 6 with -1. We have ⎡ 4 + 0 - 1 ⎤ = ⎡ 3 ⎤ = 3. Also Wednesday. Same results.

Substituting 6 with a -1 simplified the calculation. I use this shortcut a lot, because subtracting 1 is easier than adding 6. In a similar vein, any time we see a multiple of 7, we can simply drop it, because ⎡ 7 ⎤, ⎡ 14 ⎤, ⎡ 21 ⎤, ⎡28 ⎤ and ⎡ 35 ⎤ are all equivalent to 0 in mod7 world. For example:

• ⎡ 4 + 0 + 7 ⎤ = ⎡ 4 + 0 ⎤ = 4. Thursday

• ⎡ 14 + 0 + 5 ⎤ = ⎡ 0 + 5 ⎤ = 5. Friday

• ⎡ 0 + 21 + 1 ⎤ = ⎡ 0 + 1 ⎤ = 1. Monday

• ⎡ 7 + 0 + 28 ⎤ = ⎡ 0 ⎤ = 0. Sunday

These shortcuts are optional, but they do save time and reduce cognitive load.

With mod7 in our mental toolbox, we’ll make rapid progress. In the next chapter, we’ll be able to do the calendar trick with a limited scope of years and months.