GeekDad Puzzle of the Week: The Heroes of AUSM – SOLUTION

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The Affiliation of Unbelievably Superpowered Metahumans (AUSM) is trying to place its members in order of who packs the most bad-ass, city-block-wrecking, villain-squelching punch. Alphaman, Betadude,
Gammagal, Deltakid, and Epsilonimo have made various remarks – which might be true or false – about the places of themselves and others. The sentient AUSM central computer TIM has designs to terminate these jellybags and take over the planet. Its plans involve starting with the weakest one. On recording their remarks, TIM has noticed the curious fact that if the speaker was placed higher in the order than the hero or heroes mentioned, the remark was true; if the speaker was placed lower than the hero or heroes mentioned, the remark was false. When two heroes were mentioned, in no case was the speaker between them.

Since TIM cannot see which hero made which remarks, it has identified each as Hero 1, Hero 2, Hero 3, Hero 4, and Hero 5 (in no particular order). Remarks that a hero makes about him- or herself are made in the first person.

Hero 1: I was higher than Betadude and Gammagal.
Hero 2: a) Neither Alphaman nor Gammagal was fifth; b) Epsilonimo was second.
Hero 3: Gammagal was higher than Epsilonimo.
Hero 4: Deltakid was first.
Hero 5: a) Epsilonimo was fourth; b) Gammagal was lower than Alphaman.

Of course, being a superior intelligence, TIM figures out which hero was placed last in the order and begins its plans for world domination.
What are the identities of Hero 1, Hero 2, Hero 3, Hero 4, and Hero 5
and what was the order?

Solution
Set up two tables.

Ausm1Ausm1

In Table 1, you can mark off what we know about who is, or is not Hero 1, 2, 3, 4, or 5. In Table 2, you can list the information you know about placings. For example, G is not first, Hero 2 is not fourth, etc.

From Hero 1′s remark, Hero 1 is not B or G. We can’t tell whether 1 is above both B and G or below them both.

From Hero 2 statement a, 2 is not A or G. If 2a is false, then either A or G was fifth and 2a cannot be false; therefore 2a is true, neither A nor G was fifth and 2 was above both A and G; Hero 2 is not third, fourth, or fifth, and neither A nor G was first. From Hero 2 statement b, 2 is not E (can’t tell yet whether 2b is true).

From Hero 3′s statement 3 is not G or E.

From Hero 4′s statement, 4 is not D. If true, 4 would have to be above D, which is impossible; D is not first and 4 must be below D; D is not fifth and 4 is not first or second.

By marking off Table 2 accordingly, we can now see that only B and E can be first, and only B and E can be fifth; B and E between them are first and fifth, they can’t be second third or fourth.

Hero 5a is false; 5 is below E so E is not fifth; E is first and B fifth. We now know that E is Hero 1 and Hero 1 is first.

B is fifth and is Hero 2 or 5; but 2 is not fifth, therefore B is Hero 5 and is fifth. Hero 2 is D by elimination.

Hero 2 is D and D is not fourth. Hero 2 is first or second, but Hero 1 is first so 2 is second; therefore D is second.

Hero 4 is G and is not second.

Hero 3′s remark is false (E is first) so 3 is below G (as well as below E); 3 is below 4 (who is G); 3 is fourth and 4 is third ergo, A is fourth and G is third.

You will then have your tables filled in as follows:

Ausm2Ausm2

1. Epsilonimo (Hero 1)
2. Deltakid (Hero 2)
3. Gammagal (Hero 4)
4. Alphaman (Hero 3)
5 Betadude (Hero 5)

Betadude never stood a chance and Epsilonimo eventually put up a good fight but in the end, it was fruitless. All hail TIM!

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