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OK, so for those of you who think logic puzzles are just brainpower practice, I have a real-life puzzle for you as tough as any Mensa logic puzzle: creating an employee schedule! :o

A new department is created and four employees are hired to fill the department's schedule. The schedule has the following requirements:

Saturdays and Sundays - exactly one person working day shift, one person working night shift (with the other two off).

Mondays through Fridays - at least two people working day shift (which means one weekday has to be three day people, the other four weekdays will have to be two), exactly one person working night shift.

The employees have the following requirements:

Each employee must work exactly five days.

Each employee must work at least one night shift.

Each employee must have two days off in a row.

An employee can not work a day shift immediately after a night shift.

We spent about two hours on this and couldn't figure it out. It is either impossible to solve or we got stuck in a rut. Either way I figured you all would be better at this than us and can figure it out or determine that it is impossible. It really works as a classic "grid" logic puzzle - we started by creating four rows under the seven days of the week for employees A thru D, and then filling in a Saturday with Day/Night/Off/Off. We would build off of that using our rules but could never get Employee D's schedule to work.

Thanks in advance and good luck!

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Here's why:

Consider the off shifts [X]

[1] they must be taken together [XX]

[2] they must be preceded by a Night shift [NXX]. You can't have [ND].

Consider the Night shifts. [N]

[1] you need seven, so there are three [NNXX] blocks and one [NXX] block to place on the grid.

[2] they can't overlap - you need one and only one N each day.

Now the Day shifts. [D]

[1] first place the [NNXX] and [NXX] blocks so the N's don't overlap

[2] fill in the remaining spaces with D.


_ _ _ N N X X D D D N N X X D D _ _ _ <- worker A
_ _ _ D D N N X X D D D N N X X _ _ _ <- worker B
_ _ _ X D D D N N X X D D D N N _ _ _ <- worker C
_ _ _ X X D D D D N X X D D D D _ _ _ <- worker D
_ _ _ 2 3 3 3 3 3 3 2 3 3 3 3 3 _ _ _ <- shifts worked that day

Finally, assign the days.

You can't.

In any group of seven days there is only one 2-shift day, and no day where all four work.

You need two consecutive 2-shift days to cover the weekend.

In a nutshell:

Since N's are tied to X's, and you can't overlap the N's, the X's spread out and cover all 7 days.

Thus you can never have all four working on the same day.

Advice: Let someone work a Day shift after a Night shift, and pay her time and a half.

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