Hirofumi Nakai Improved (HNI) - a method for Doomsday calculation

Hirofumi-Nakai-Improved.md

Hirofumi Nakai Improved

The Doomsday rule was devised by John Conway in 1973 to compute the day of week in your head (https://en.wikipedia.org/wiki/Doomsday_rule). Others have made improvements for easier mental calculation.

Currently, the simplest one is published in 2023 in the paper "A Simple Formula for Doomsday" by Hirofumi Nakai (https://thatsmaths.com/2023/06/22/a-simple-formula-for-the-weekday/).

• split the year into century part and the remaining 2-digit part: Y = 100c + y
• compute the modulo 4 of these 2 parts: c₄ = c % 4, y₄ = y % 4
• the Doomsday would be (5(c₄ + y₄ - 1) + 10y) % 7

I made some improvements to make it even easier.

(5(c₄ + y₄ - 1) + 10y) % 7

= (5(c₄ + y₄ - 1) + 10y - 14y - 7(c₄ + y₄ - 1)) % 7

= (-2(c₄ + y₄ - 1) - 4y) % 7

= 2(1 - c₄ - y₄ - 2y) % 7

Since y has 2 digits y = 10a + b, we can quickly compute 2y % 7

2y % 7

= (20a + 2b) % 7

= (20a + 2b - 21a) % 7

= (2b - a) % 7

Replace 2y with this, we have a new formula which I named HNI (Hirofumi Nakai Improved)

2(1 - c₄ - y₄ + a - 2b) % 7

Faster mental math

The above formulas are simple, but not so quick. If you want to calculate faster, you should memorize some values. For more detail, see https://worldmentalcalculation.com/how-to-calculate-calendar-dates/

The calculation of Doomsday contains 2 parts: W = (C + Y) % 7

• C is based on the century part: C = (2 - 2c₄) % 7
• Y is based on the year part: Y = 2(a - 2b - y₄) % 7

You can memorize the values for C = m2053(c₄) = (2 - 2c₄) % 7:

c₄	0	1	2	3
(2-2c₄)%7	2	0	5	3

Don't worry! There are only 4 alternating values for C:

• C=2 with year 1600s / 2000s / 2400s …
• C=0 with year 1700s / 2100s / 2500s …
• C=5 with year 1800s / 2200s / 2600s …
• C=3 with year 1900s / 2300s / 2700s …

The remaining part you have to calculate is Y = 2(a - 2b - y₄) % 7. These numbers are quite small, so you can do it quickly with some practise.

You can also add or subtract 7/14/21/... to make intermediate value even smaller.

Using the notation b₄ = b % 4, we can make it faster:

• if a is even, y₄ = b₄ = (b-4)₄ = (b-8)₄ (you can subtract 4 or 8 instead of dividing by 4)
• if a is odd, y₄ = (b+2)₄ = (b-2)₄ = (b-6)₄ (you can subtract 2 or 6 instead of dividing by 4)

Even faster with memorization

Y = 2(a - 2b - y₄) % 7

= (2a - 4b - 2y₄) % 7

= (2a + 3b - 2y₄) % 7

= (2a + b + 2(b - y₄)) % 7

when a is even, 10a % 4 = 0, thus y₄ = b₄

= (2a + b + 2(b - b₄)) % 7

= (2a + b + 2(4(b/4))) % 7

= (2a + b + b/4) % 7

when a is odd, 10a % 4 = 2, thus y₄ = (b+2)₄

= (2a + b + 2(b - (b+2)₄)) % 7

= (2a + b + 2(b+2 - (b+2)₄) - 4) % 7

= (2a + b + 2(4((b+2)/4)) - 4) % 7

= (2a + b + (b+2)/4 - 4) % 7

= (2a + b + (b+10)/4 - 6) % 7

= (2a + b + (10+b)/4 + 1) % 7

Now, Y requires only simple math with a and b, the 2 digits of y.

You can also memorize values of b/4 and (10+b)/4+1 to save time.

b	0..3	4..7	8,9
b/4	0	1	2

b	0,1	2..5	6..9
10+b	10,11	12..15	16..19
(10+b)/4+1	3	4	5

Instead of memorization, we can simplify the formula

when a is odd (using p as the 9-complement of b p = 9 - b)

= (2a + b + (b+2)/4 - 4) % 7

= (2a + b - 2 + (b+2-8)/4) % 7

= (2a + b - 9 + (b-9+3)/4) % 7

= (2a - (9-b) + (-p+3)/4) % 7

= (2a - p - p/4) % 7

This is similar to the formula when a is even (with -p in the place of 'b')

Another formula for Y

Y = (y + y/4) % 7

= (10a + b + (10a+b)/4) % 7

= (3a + b + 2a + (2a+b)/4) % 7

= (b - 2a + (b+2a)/4) % 7

= (b - a + (b-2a)/4) % 7

if you don't mind some multiplication, we can make it shorter:

= ((5b - 6a)/4) % 7

= ((5(b-a) - a)/4) % 7

Yet another formula for Y

Y = (y + y/4) % 7

= (y₄ + 4(y/4) + (y/4)) % 7

= (y₄ - 2(y/4)) % 7

= (y₄ - 4(y/4)/2) % 7

= (y₄ - (y-y₄)/2) % 7

= (y₄ - (10u+v)/2) % 7

= (y₄ - 5u - v/2) % 7

= (y₄ + 2u - v/2) % 7

Here we assumed that that y-y₄ = 10u + v (v is even because (y-y₄) % 4 = 0)

Further reading

Thanks to Miroslav (@Borg19l71) for sharing prior art research. There are many other formulas. You can see more at https://qr.ae/pCX9gy

Memorization for all parts of date (day, month, year, century) https://davecturner.github.io/2021/12/27/doomsday-rule.html

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=2(1 - c₂ - y₂ + u - 2v) (span 66)(6 steps) can be written as

=2(u+v+c₂'+y₂')+v

(span 57)(6 steps) where c₂',y₂' are analogs of negative moduli where counting is upward and not downward like in common modulo operation.
(only one multiplication, without subtraction, no negative numbers)
Analogs of negative modulo:
Instead of normal modulo4=m4=0123 (count downward 0..3 steps until number divisible by 4)
analog of negative modulo4 gives=4-m4=4321 (count upward 1..4 steps (at least one step) until number divisible by 4)
Span is 2x(2x(9+3))+9=4x12+9=48+9=57
Example: 1979=> 2(7+9+1+1)+9=2*(18-14)+9=24+9=17=17-14=3
Example: 2000=> 2(0+0+4+4)+0=28=16=16-14=2
https://qr.ae/pCX9gy (one of my contributions)
-c2,-y2 can be calculated as "opposite modulo"=4's complement of modulo by counting upward instead downward like for common modulo
-2v can be accounted by calculating with 9's complement of v: -2v<=>2v'=2(9-v) and changing anchor value:
f(c, y)=2(u+2v'+c2'+y2')-1 where v'=9-v
v',c2',y2'=analogs of negatives of v,c2,y2 (just count upward instead of downward)
For analog of negative modulo: count at least one! (c2',y2' must yield 1..4)
Example: Analog of -11%4=-3 is counting from 11 upward (at least once): 12=>once=1
It's not equal, only analog=equal to a constant shift
Anchor value (like for year 2000) is 2(0+29+4+4)-1=226-1=51=51-49=2 (for Doomsday method)
Span is 2*(9+29+3+3)=2(27+6)=66
(Function with values ranging from min to max has span=max-min)
(Span is the measure of result spreading interval which is loosely correlated with complexity of function)
(mod7 operations are not included in calculation of span)
Common part of formula is century term
f(c)=2-c2=2+2c2'=2(1+c2')=2053(c2) (function of c2 with values 2,0,5,3)
Other part is f(y): (some examples) (there are more than 20 formulas)
f(y)=
=2(u-2v-y2) span 60 steps 5
=2(u+2v'+y2') span 60 steps 5
=y2-(y-y2)/2 span 51 steps 4
=y2+[2u-v/2](y-y2) span 24 steps 4 where [2u-v/2] is function of y-y2
=2u-v/2+0134(y2) span 26
which can be memorized with 2x10 tables
0246 1350 24 or 02461 35024
0066 5544 33 or 00665 54433
and mnemonic 0134 for y mod 4 part
(and mnemonic 2053 for c mod 4 part)
(custom anchor values can be incorporated by changing these short mnemonics if needed)
=2u-v/2+y2+y2/2 span 26
Simplest formulas for year in century part have 4..5 steps with shortest span =15 (does not mean it is the simplest)
f(y)=2u+v+00001111223344445555(y%20) span 32
f(y)=x+f(n4(x)) span 9 steps 4
where x=y%28 n4(x)=x-x%4=4n f(4n)=0,+1,-5,-11,-10,-16,-22,-28,-27 n=0..8
and %28=do nothing or -30+2 or -60+4 or -90+6

thanks, @Borg19l71 I included my interpretation of your formulas into my gist.

The following is the answer of Miroslav Janković on 2025-08-28
(copied from https://qr.ae/pCX9gy):

date= year-m-d :

year=xccab=10000x+100cc+10a+b

x=don’t care ( Gregorian calendar repeats every 10000 years)

———————————————————————————————————————

Weekday W=C+Y+d+(M+L) for monthly positive methods like

Weekday W=C+Y+d-(M+L) for monthly negative methods ( AKA Doomsday methods )

Weekday W=W%7=W±7n ( modulo 7 )

1=1st WORKDAY=Monday 2=2nd WORKDAY=Tuesday etc. 6=Saturday 7=0=Sunday

———————————————————————————————————————

century number = year/100=xcc or (year%10000)/100=cc

century number xcc or cc is not equal to classic century.

————————————————————————————————————————————-

number/code C for century number:

C= A–2*(cc%4) ( for Gregorian calendar dates ) ( A is anchor value for given method ) or

C=A+cc/4–2cc ( Zeller’s Congruence method for Gregorian calendar dates ) and

C = (26+A)–cc ( for Julian calendar dates ) (number 26 can be replaced with 26-7z=26,19,12,5 )

Usually mnemonics are given from century number divisible by 4 e.g. century number 0..3

Gregorian century table for Standard method C=6420 (A=6) Julian C=6543210 6543210 65(43210)

Gregorian century table for Doomsday method C=2053 (A=2) Julian C=2106543 2106543 21(06543)

————————————————————————————————————————————-

number/code Y=f(y) for year in century y=10a+b=00..99:

( a=year tens )

( b=year ones )

( many methods )

————————————————————————————————————————

Y=f(y)=

1) (5y)/4

2) y+y/4 ( As in Zeller’s Congruence method )

3) (y%7)+y/4

4) q+f(rest) where q=y/12 rest=y-12q=y%12

5) (y%4)-rest/2=(y%4)+2a-b/2 where rest=y-(y%4)=4n=10a+b

6) odd+11 method=-(y+11 until divisible by 4)/2:

6a) if (y) odd +11

6b) divide by 2

6c) if odd +11

6d) negate (by modulo 7)=count upward to number divisible by 7=7n

7) 2a+b+G where G=(b+14(a%2))/4=group number where

G=0,1,2 for years g=0..3, 4..7, 8..9 for even decades

G=3,4,5 for years g=10..11, 12..15, 16..19 for odd decades

where g=(y%20) year in a group of 20

8) 2a[+3 if a odd]+b+b/4[+1 if a odd and b=2,3,6,7]

9) 2a

+(b+b/4) if a even

-(c+c/4) if a odd where c=9-b ( 9’s complement of b )

10) b-a+floor((b-2a)/4)

11) floor((5b-6a)/4)

12) f(base)+(y-base)+leap_years_from_base_plus_1_upward_to_y_inclusive

where base years can be selected so f(base)=0:

base=00,06,17,23,28,34,45,51,56,62,73,79,84,90

13) mod28 and minus method:

13a) mod28 <=> ( -30+2 or -60+4 or -90+6 )

13b)

if 4.. => +1

if 8.. => -5

if 12.. => -11

if 16.. => -10

if 20.. => -16

if 24.. => -22

if 28.. => -28

if 32.. => -27

14) odd-3 method:

14a) if odd -3

14b) divide by 2

14c) if oddness changed from start -3

14d) negate (by modulo 7)=count upward to number divisible by 7=7n

15) -2(2y+y%4) or -2(2(y%7)+y%4) or 3y-2(y%4) or 3(y%7)-2(y%4) or -2(2b-a+y%4) or 2a+3b-2(y%4)

16) Zeller’s congruence method:

Y+C=(y+y/4)+(A+cc/4–2cc) ( Gregorian calendar dates ) or

Y+C=A+year+year/4+year/400–year/100 ( Gregorian calendar dates ) and

Y+C=(y+y/4)+(26+A-cc) ( Julian calendar dates ) or

Y+C=5+A+year+year/4 ( Julian calendar dates )

17) Y+C=A+4(year%100)-(year%400)-2(year%4) ( origin of method 15) )

———————————————————————————

number/code M for month m = number of odd days before that month ( standard monthly positive method )

number of odd days for months are 303 232 332 323 ( subtract 28 from month length in days )

( counting only for non-leap year )

So for Jan=0 ( because 0 days )

for Feb=Jan+odd_days_in_Jan=0+3=3

for Mar=Feb+odd_days_in_Feb=3+0=3

for Apr=Mar+odd_days_in_Mar=3+3=6

for May=Apr+odd_days_in_Apr=6+2=8=8–7=1 ( by modulo 7 )

for Jun=May+odd_days_in_May=1+3=4

for Jul=Jun+odd_days_in_Jun=4+2=6

for Aug=Jul+odd_days_in_Jul=6+3=9=9–7=2 ( by modulo 7 )

for Sep=Aug+odd_days_in_Aug=2+3=5

for Oct=Sep+odd_days_in_Sep=5+2=7=7–7=0( by modulo 7 )

for Nov=Oct+odd_days_in_Oct=0+3=3

for Dec=Nov+odd_days_in_Nov=3+2=5

so month table M=033 614 625 035 ( standard method )=M0

——————————————————————————————————————

Doomsday method has values M=3-M0-L ( negative monthly method )

(Leap year correction included in the number for month M)

——————————————————————————

L=leap year correction=-1 for Jan/Feb of leap year else 0

——————————————————————————

proof: 3-M0-L= 3–0-(0/-1), 3–3-(0/-1), 3–3 3–6,3–1,3–4, 3–6,3–2,3–5, 3–0,3–3,3–5

=3/4, 0/1, 0, -3,2,-1, -3,1,-2, 3,0,-2

=3/4, 28/29, 0/14, 4,9,6, 11,8,5, 10,7,12

which is the same because differences are ±7n

——————————————————————————————————————-

example 2053-June-30th : ( Doomsday method ) :

20xx—->2–2*(20%4)=2–2*(20–5*4)=2–2*(20–20)=2–2*0=2–0=2

June 6/6 => 6

Day in month=30=30–7n=30–28=2

————————————————-

year in century=53:

1) (5*53)/4=265/4=66

2) 53+53/4=53+13=66

3) 53%7+53/4=(53–49)+13=4+13=17

4) 48/12+(53–48)+(53–48)/4=4+5+5/4=9+1=10

5) =1-(53–1)/2=1–52/2=1–26=-25

6) -(53+11)/2=-64/2=-32 or

6a) 53+11=64

6b) 64/2=32

6d) from 32 to 35=3

7) 2*5+3+(G=4 for g=13)=17

8) 2*5+3+3+3/4+1=17

9) 2*5-(6+6/4)=10–7=3 because 9–3=6

10) 3–5+floor((3–2*5)/4)=-2+floor(-7/4)=-2–8/4=-2–2=-4

11) floor((5*3–6*5)/4)=floor((15–30)/4)=floor(-15/4)=-16/4=-4

12) (53–51)+1=3

13a) 53–30+2=25

13b) 25–22=3

14a) 53-3=50

14b) 50/2=25

14d) -25

15) -2(2(y%7)+y%4)=-2(2(53–49)+(53–52))=-2(2*4+1)=-2(8+1)=-2*9=-18

or -2(2b-a+y%4)=-2(2*3–5+(53–52))=-2(1+1)=-2*2=-4

or 2a+3b-2(y%4)=2*5+3*3-2(53-52)=10+9-2*1=19-2=17

(All methods give same result because differences are ±7n)

16) Y+C=(53+53/4)+(A+20/4–2*20)=66+A+5–40=A+31 or

Y+C=A+2053+2053/4+2053/400–2053/100=A+2053+513+5–20=A+2551

( difference is 2551–31=2520=7*360 )

17) Y+C=A+4*53-53-2*(53-52)=A+3*53-2*1=A+159-2=A+157

———————————————————————————————

So 2053.June 30th:

W=2+3+2–6=1=Monday (Doomsday method)

W=6+3+2+4=15=15–14=1=Monday (Standard method)