The open-day formula with rest-day sets: five weekend conventions and cross-border spans

The formula for counting open days does not change from country to country. What changes is one of its arguments, and hard-coding that argument is what makes a calculator wrong outside the place it was written.

This page computes nothing. It gives the general form, the figures it produces under several conventions, and the harder case where the two ends of a span sit under different ones.

What to take away

The general form

Let R be the set of rest-day names and k its size. For a span of n days beginning on weekday s:

open(s, n, R) = (7 - k) * floor(n / 7)
              + count of the n mod 7 days from s whose name is not in R

Nothing in that refers to Saturday. Substituting R = {Sat, Sun} gives the five-day version, R = {Fri, Sat} gives a common alternative, and R = {Sun} gives a six-day week. The whole-weeks term is fixed by k alone, and only the remainder term looks at the start day.

Because the remainder is at most six days, the difference between two conventions over a long span is bounded by the difference in k times the number of weeks, plus at most six. That bound is what tells you when a discrepancy is a convention difference and when it is a mistake.

The same spans, several conventions

All in 2027, closures ignored so that only the weekend rule is in play.

Three things are visible there. Any two-rest-day convention gives the same answer for a whole month, because a month is four whole weeks plus a remainder, and the remainder rarely distinguishes them. A short span can distinguish them sharply: the Friday to Monday row is 2 under a Saturday and Sunday weekend and 3 under a Thursday and Friday one.

A one-rest-day convention differs from a two-rest-day one by roughly a day a week, which over a year is 52.

The annual totals show the same pattern: 261 for a Saturday and Sunday weekend in 2027, 260 for Friday and Saturday, and 313 for Sunday alone. The derivation of those annual figures is under the annual formula, and the monthly version is under counting a month.

When the span has an end in each country

Two parties in different countries counting the same deadline are not applying one formula with a disputed argument. They are applying two formulas, and the honest description is two answers.

Three arrangements are used in practice, and each is a decision rather than a calculation.

One governing calendar. The contract names a country and everyone counts under it. This is the only arrangement that produces a single number nobody has to reconcile, and the number will sometimes look wrong to the party in the other country.

The intersection of open days. A day counts only if it is open in both places. Under a Saturday and Sunday convention on one side and a Friday and Saturday one on the other, the rest set becomes Friday, Saturday and Sunday, and March 2027 drops from 23 to 19. This is the conservative reading and the one to use when both parties must act on the same day.

The union of open days. A day counts if it is open in either place. Under the same pair the rest set shrinks to Saturday alone, and March 2027 rises to 27. This suits a hand-off where either party can progress the work.

State which of the three applies before anyone counts. The difference between the intersection and the union over one month is eight days, which is larger than most closure lists.

Why the argument is a set and not a number

A calculator that takes "days in the working week" as a number cannot express which days those are, so it cannot compute a short span correctly and cannot answer the intersection question at all. The set is the minimum honest input.

Represent it as seven flags read from Monday to Sunday, which is compact, sorts, and can be compared between two calendars without parsing anything. Where those conventions come from, and how a national list of closures is published on top of them, is under how closure calendars are made.

The variety itself is not exotic. The history of how a rest day came to be one day, then two, in different places is under workweek and weekend, and the notion of a working day as a legal category is under business day.

The layer below a country, where a region adds its own closures on top of the national list, is under state and regional calendars, and the day-stepping form of the same arithmetic is under adding business days.

Common questions

Does the formula change for a country with a half-day rest convention?

Not at day resolution, because a half day is not a rest day. It has to be handled at hour resolution, where the day is a shorter window.

Why do two-rest-day conventions agree on a whole month but not on a short span?

Because a month is four whole weeks plus at most three days, and the whole weeks contribute the same amount under any two-rest-day set. Only the leftover days can distinguish them.

Can I convert a count from one convention to another?

Not reliably. The difference depends on the start day, so a conversion factor is right on average and wrong on most individual spans.

Which arrangement should a contract use for a cross-border deadline?

Whichever one matches what actually has to happen on the day, and it should say so in words rather than leaving it to whoever runs the calculation.