Inverse Doomsday

[Thomas Klemm]

Recently I came across an old concert poster that took place on Friday, September 4th and wondered what year this was.
After a few tries with the DOW (day of week) function I figured it out pretty quickly.
But then I wondered how to solve this problem without brute force.

In this specific case, the closest Doomsday is September 5th (i.e. 9/5) which is a Saturday and therefore the anchor day of this year which is coded as 6.

For the year's last two digits \(y\) in a certain century, its anchor day can be calculated using the following formula:

\(d=\left(k + y+\left\lfloor {\frac {y}{4}}\right\rfloor \right){\bmod {7}}\)

The constant \(k\) depends on the century:
\(1900–1999\): \(k = 3\)
\(2000–2099\): \(k = 2\)

We therefore essentially want to invert the following function:

\(m=\left(y+\left\lfloor {\frac {y}{4}}\right\rfloor \right){\bmod {7}}\)

This function is periodic with a period of \(28 = 4 \times 7\).
We will therefore limit ourselves to \(y \in \{0, 1, 2, \cdots, 27\}\) for the time being.

The following table can be viewed as a torus.
It indicates for each row which \(y\) are mapped to the same value \(m\):

\(
\begin{array}{|r:r|r|r|r|}
\hline
& 0 & 1 & 2 & 3 \\\hline
0 & 0 & 17 & 6 & 23 \\\hline
1 & 12 & 1 & 18 & 7 \\\hline
2 & 24 & 13 & 2 & 19 \\\hline
3 & 8 & 25 & 14 & 3 \\\hline
4 & 20 & 9 & 26 & 15 \\\hline
5 & 4 & 21 & 10 & 27 \\\hline
6 & 16 & 5 & 22 & 11 \\\hline
\end{array}
\)

A row is skipped if the year is a leap year, which means it is divisible by \(4\).

In my case (i.e. Friday, September 4th in the 20th century) we are looking for values \(y\) that map to \(3 = 6 - 3\).
This means that we consider the following line:

\(
\begin{array}{|r:r|r|r|r|}
\hline
3 & 8 & 25 & 14 & 3 \\\hline
\end{array}
\)

Or then in correct order: \(\{3, 8, 14, 25\}\).
Once the years are sorted in each row you may notice a common pattern:
The differences cycle through the following values: \(\{5, 6, 11, 6\}\)
However, they will start with different values.

This program is for the HP-42S:

Code:

00 { 59-Byte Prgm }
01▸LBL "DD↑-1"
02 GTO IND ST X
03▸LBL 00
04 GTO 09
05▸LBL 01
06 GTO 07
07▸LBL 02
08 GTO 10
09▸LBL 03
10 GTO 08
11▸LBL 04
12 9
13 GTO 07
14▸LBL 05
15 4
16 GTO 09
17▸LBL 06
18 5
19▸LBL 07
20 STOP
21 6
22 +
23▸LBL 08
24 STOP
25 5
26 +
27▸LBL 09
28 STOP
29 6
30 +
31▸LBL 10
32 STOP
33 11
34 +
35 GTO 07
36 END

Example

3 XEQ "DD↑-1"

3

R/S

8

R/S

14

R/S

25

R/S

31

\(\cdots\)

81

R/S

87

R/S

92

R/S

98


Can you come up with a simpler solution?

PS: Due to further constraints, I am convinced that the concert took place in 1992.

References

• Doomsday Rule