# Who's that Simple Pokémon™?

For every alphabet (A to Z), there exists at most one corresponding Simple Pokémon™. As of Generation 8, it goes:

Ariados
Blaziken
Charmeleon
Diggersby
Ekans
Farfetch'd
Groudon
Hypno
Ivysaur
Jellicent
Koffing
Lapras
Metapod
Nidoran♀
Oddish
Poliwag
Qwilfish
Rattata
Squirtle
Tentacruel
Umbreon
Venusaur
Wynaut
(currently none for X)
Yanma


What is the Simple Pokémon™ that starts with Z?

Subtle Hint:

If it was not for Poliwag, it would be Patrat instead.

Moderate Hint:

This list would be updated if Pokémon #1092 is revealed.

Decisive Hint:

Association closed: Inverted identity.

Decisive Hint 2:

The decisive hint clues to a 5-letter word.

• Can you define a Simple Pokémon™? Is it a non-legendary or non-pseudolegendary? Apr 17 '20 at 1:19
• @Soham Konar "Simple" doesn't mean it's not legendary. There is Groudon in the list. Apr 17 '20 at 2:33
• @SohamKonar Working out the definition of "a Simple Pokémon™" from the list given is step 1 of the puzzle. Step 2 is identifying a Pokémon matching that definition whose name starts with the letter "Z" Apr 17 '20 at 12:01
• This is going to bug me for a very long time. Apr 17 '20 at 14:27
• A clarification about the moderate hint: if the next Pokémon generation add less than 201 new Pokémon will the moderate hint still be valid? Apr 19 '20 at 10:13

TL;DR

Zubat (pokédex number 41).

Long explanation:

The (national) Pokédex number of the Simple Pokémon™ are: 168, 257, 5, 160, 23, 83, 383, 97, 2, 593, 109, 131, 11, 29, 43, 60, 211, 19, 7, 73, 197, 3, 360, 193

That are

Prime numbers except for 660 (Diggersby), 360 (wynaut) 168 (ariados), and 60 (poliwag);

Those four exceptions and the ones stated in the Hints are (tanks to @Will):

So the Simple Pokémon™ associated to each letter is (thanks @Stiv) the Pokémon which name starts with that letter and

with lowest national Pokédex number equivalent to the order of a non-cyclic simple group. If no such Pokémon exists, the Simple Pokémon for a letter is the Pokémon with lowest prime national Pokédex number.
We fall back on prime numbers because they are the orders of cyclic simple groups; the listed Pokémons are "Simple" because the definition involves simple groups.

So the Simple Pokémon™ starting with Z is

Zubat (pokédex number 41)

since

- there is no Pokémon starting with Z which Pokédex number is the order of a non-cyclic simple group; and
- among all the Pokémon starting with Z 41 is the lowest prime Pokedex number.

A note about the Moderate Hint:

It could be the case that this list is not updated even when the Pokémon #1092 is revealed, for example if its name starts with P (for the same reason explained in the subtle hint). Just wait and we will see!

• This looks like you're on to something - note that rot13(Bqqvfu vf 43), so that actually fits with your hypothesis. In general, these all seem to be rot13(gur svefg Cbxrzba ortvaavat jvgu gung yrggre gb unir n cevzr Cbxrqrk ahzore), with the 2 other exceptions you list as cases to puzzle over... This also explains why no X (Xatu=178, Xerneas=716, Xurkitree=796...).
– Stiv
Apr 17 '20 at 15:18
• @Stiv thank you for the reply. I actually meant Poliwag instead of oddish: I copied the wrong number since they are one next to the other. I was thinking about some constraint for which there could be "at most one Simple Pokémon™" starting with that letter. rot13(gur svefg bar gung...) is a good constraint indeed, updating my answer. Apr 17 '20 at 15:54
• Interesting why Ariados is there instead of Arcanine, by this logic. Or Poliwag instead of Pidgeotto. Or Wynaut instead of Weedle. These three Pokémon I’d say fit your pattern better than OP’s do.... @melfnt Apr 17 '20 at 15:57
• Interesting that the numbers of noted exceptions (including hints) are the first terms here?
– Will
Apr 26 '20 at 21:28
• I don't know much of the mathematics involved in this puzzle either but it's actually the case that rot13(plpyvp fvzcyr tebhcf unir beqref gung ner nyy gur cevzr ahzoref). So I think that your new formulation is ok but it should state that rot13 (jr snyy onpx ba cevzrf orpnhfr gurl'er gur beqref bs plpyvp fvzcyr tebhcf)
– Will
Apr 27 '20 at 12:50