# French Public Holidays

We consider the following rules that apply to the 11 French public holidays :

1. New Year's Day : 1st of January
2. Easter Monday : Monday following Easter Sunday which is the next Sunday after the first full moon beginning from Spring's Equinox (approximately 21 march).
3. Labour Day : 1st of May
4. Victory in Europe Day : 8th of May
5. Ascension Day : 39 days after Easter Sunday
6. Whit Monday : 50 days after Easter Sunday
7. Bastille Day : 14th of July
8. Assumption of Mary to Heaven : 15th of August
9. All Saints' Day : 1st of November
10. Armistice Day : 11th of November
11. Christmas Day : 25th of December

# Date distance

We here define a pseudo-metric on dates, that we will now on call a distance to simplify for everyone $$d:D\times D \longrightarrow \mathbb Z$$ where:

• $$D$$ is the set of all the possible dates.
• $$D=\{aaaa/mm/dd\}$$
• $$aaaa$$ is a year, typically $$2020$$ or $$1996$$
• $$mm$$ is a month, typically $$04$$ for April
• $$dd$$ is a day
• $$d$$ will be closely linked to the French Public Holidays in the sense that it counts the number of French Public Holidays between two dates.
• For instance: $$d(2020/12/24, 2020/12/26) = 1$$ because there is exactly one public holidays between those two dates: Christmas Day.
• Note that $$d(2020/12/25,2021/01/01) = 0$$ (both bounds are excluded)
• A last example is $$d(2020/04/10, 2020/05/10) = 3$$

# Puzzle

Let us suppose you have $$12$$ dates to define $$\forall (i,j) \in \{1,2,3\}\times \{1,2,3,4\},\quad d_{ij}\in D$$. With the constraint that $$d_{ij} \ge 2005/11/25$$ (this is a symbolic and special day for me) and $$d_{ij} \le$$ the day this puzzle is posted.

We thus define :

• $$r_i = d(d_{i1}, d_{i2}) + d(d_{i2}, d_{i3}) + d(d_{i3}, d_{i4})$$
• $$\displaystyle c_j = d(\min_i{d_{ij}}, \max_i{d_{ij}})$$

$$\begin{array}{rrrr|r} c_1 & c_2 & c_3 & c_4 & \\ \hline d_{11} & d_{12} & d_{13} & d_{14} & r_{1} \\ d_{21} & d_{22} & d_{23} & d_{24} & r_{2} \\ d_{31} & d_{32} & d_{33} & d_{34} & r_{3}\\ \end{array}$$

This calculation puzzle is to compute

### $$\max z = r_1 + r_2 + r_3 - c_1 - c_2 - c_3 - c_4 + w$$

$$\displaystyle w = \sum_{i=1}^3\sum_{i'=1, i'\neq i}^3\sum_{j=1}^4\sum_{j'=1, j'\neq j}^4 d(d_{ij},d_{i'j'})$$

with all the previous constraint and the final one that:

• $$d_{ij} \neq d_{i'j'} \text{ if } i \neq i' \text{ or } j \neq j'$$ (all $$d_{ij}$$ are unique)

Computers are allowed :)

• Isn't $d(2020/04/10, 2020/05/10) = 3$? Easter Monday, 1/5, 8/5. – Culver Kwan Apr 10 at 7:44
• @CulverKwan Oh yeah! You're 100% right, thanks! – JKHA Apr 10 at 7:47

I think that

$$\max z = 8892$$

This can be achieved with the following choices

$$(d_{11}, d_{12}, d_{13}, d_{14})$$ = (2005/11/25, 2020/04/05, 2005/11/28, 2020/04/08)
$$(d_{21}, d_{22}, d_{23}, d_{24})$$ = (2005/11/26, 2020/04/06, 2005/11/29, 2020/04/09)
$$(d_{31}, d_{32}, d_{33}, d_{34})$$ = (2005/11/27, 2020/04/07, 2005/11/30, 2020/04/10)

Reasoning

First we assume that no $$d_{ij}$$ coincides with a holiday, we can only lose from the value of $$z$$ by doing so. Also, we can set each $$c_j = 0$$ as the distances between dates in the same column only contribute negatively to $$z$$. For convenience we will pick the dates in each column to be consecutive.
Under these assumptions $$z$$ equivalent to $$3 \left[ 5d(d_{11}, d_{12}) + 5d(d_{12}, d_{13}) + 5d(d_{13}, d_{14}) + 4d(d_{11}, d_{13}) + 4d(d_{11}, d_{14}) + 4d(d_{12}, d_{14} )\right]$$ This is a slightly asymmetric expression so it makes most sense to maximise the parts with coefficient $$5$$. This maximum is $$156$$ in each case, achieved by putting dates at opposite ends of the date range. Then, we also get the additional $$d(d_{11},d_{14})=156$$ term for free (although the other terms are zero).
This makes the maximum at $$3 \times 19 \times 156 = 8892$$

• I think your objective is wrong, especially for the term $w$! $w$ is the sum of all the distances between each couple of $d_{ij}$ and $d_{i'j'}$, those being different. But I +1 you anyway because there are great ideas in you answer – JKHA Apr 13 at 18:45
• @JKHA Your last comment is incorrect. w is the sum of all distances between elements which are in a different row and column to each other. Check your summation again, both i' and j' are different to i and j. – hexomino Apr 14 at 15:30
• Oh indeed! I wrote something different than I wanted. Well, I'm not changing my puzzle now it has an answer, congrats ;) – JKHA Apr 15 at 0:08