GeekDad Puzzle of the Week: The AUSM Holiday Schedule – Solution

Congratulations and a Happy Boxing Day to Steve Benkovic, who correctly scheduled the heroes of AUSM and receives a $50 gift code to ThinkGeek! Check the solution after the jump for $10 off of your next ThinkGeek purchase of $30 or more.

Special thanks to ThinkGeek for providing our prizes!

The holiday schedule kicks in at the headquarters of the Affiliation of Unbelievably Superpowered Metahumans (AUSM). Maintenance, IT, HR, all must rearrange their work to accommodate holiday time off and such.
The heroes of AUSM, Alphaman, Betadude, Gammagal, Deltakid, and
Epsilonimo, realizing that evil does not take a holiday, must still hold monitor duty. However, they cannot agree on a new schedule. They come up with the following rules:

1. If Alphaman is present, Betadude must be absent unless Epsilonimo is absent, in which case Betadude must be present and Gammagal must be absent.

2. Alphaman and Gammagal may not be present together or absent together.

3. If Epsilonimo is present, Deltakid must be absent.

4. If Betadude is absent, Epsilonimo must be present unless Gammagal is present, in which case Epsilonimo must be absent and Deltakid must be present.

To be fair, the heroes have agreed that there should be a different set of heroes present at the monitors on each of the next seven days.

How do the heroes solve their problem? What should the different sets of those present and absent be while conforming to the rules they have established?

SOLUTION

From the first rule, with A present the arrangement must include:

a) Present:  A    Absent: B
or
b) Present:   AB    Absent: GE

From the second rule, G is absent in (a). From rule one and three, E is present and D absent in (a). In (b) D may be present or absent so there are three possibilities with A present:

(a) Present: AE    Absent: BGD
(b) Present: ABD    Absent: GE
(c) Present: AB    Absent: GDE

Per rule four, another arrangement might be: Present: GD; Absent: BE. Per rule two, A must be absent in this so:

(d) Present: GD        Absent: ABE

Consider other arrangements with A absent and we thus have G present. Suppose E is present and therefore D is absent. From rule four we can’t have B absent but we can have B present, giving another arrangement:

(e) Present: BGE        Absent: AD

Suppose with G present and A absent we have E absent. If we thus have B absent then rule four dictates we have D present and we are back to option (d). But suppose B is present, then D may be present or absent:

(f) Present: BGD        Absent: AE
(g) Present: BG        Absent: ADE

All possible permutations are exhausted. The seven different arrangements are as follows:

Day 1
Present: AE        Absent: BGD

Day 2
Present: ABD    Absent: GE

Day 3
Present: AB        Absent: GDE

Day 4
Present: GD        Absent: ABE

Day 5
Present: BGE    Absent: AD

Day 6
Present: BGD    Absent: AE

Day 7
Present: BG        Absent: ADE

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