Solution: Missing Members

To solve this puzzle, you need to realize three things:

• Each color corresponds to some list of things, with an arrow from each thing to the next in the list.
• Each node corresponds to a first letter (so two things will be at the same node if they start with the same first letter).
• Each list has one item missing, which is just skipped (so there's an arrow from the item before it to the one after it).

This is somewhat hard, but there are several things that confirm each piece:

• The emojis at the bottom, "️", clue the lists to use. Experienced teams will often be able to quickly figure out from this not just that lists are relevant, but the exact lists to use.
• The title clues "missing" very strongly, and sort of makes more sense if the "members" are members of some list.
• There are 19 nodes, which is slightly less than 26. There's way too much overlap for each node to be a word (not many words are part of multiple lists, or part of the same list multiple times), but each node being a letter and a few letters being unused makes sense.
• Counting the arrows of each color gives 11, 8, 10, 5, 10, 11. (Only looking at one color at a time makes this a lot easier.) If you've figured out the lists to use, these numbers are each two less than the number of items in a list, meaning that there are arrows between consecutive items (leading to (items - 1) arrows) but with one item removed (leading to (items - 2) arrows instead).

Once teams figure out the three things above (or even just the first two), it's natural to try to figure out the category corresponding to each color and the letter corresponding to each node. From the emojis at the bottom, we can figure out (with a few possible red herrings) that the categories are playing card names, first 13 states (by order of admission), musical notes, months, Chinese zodiac animals, and rocks on the Mohs scale. However, this is clearly not the red-purple color order, so we will have to do some experimentation to match them up. (Note: The emojis at the bottom are (on expert track) instead ordered by first set item: ace, delaware, do, january, rat, talc.)

If teams haven't done counting the arrows of each color yet, doing it now will narrow every category down to the one or two options with the right number of arrows. Another very helpful thing is nodes that feed back to themselves; those have to be two consecutive items starting with the same letter (or two items with one in between, if the one in between is the missing one). For example, the Chinese zodiac can't have a node that feeds back to itself, so if you're matching blue and yellow (both with 10 arrows, so 12-item categories) to months and the Chinese zodiac, you can determine blue is months. More intricately, we can figure out that the first 13 states can't have three nodes feeding back to themselves, so purple has to be face cards.

The resulting matching is:

Color	Category	Number of items	Number of arrows	Emoji
Red	First 13 states	13	11
Orange	Mohs scale	10	8
Yellow	Chinese zodiac	12	10	️
Green	Musical notes	7	5
Blue	Months	12	10
Purple	Playing cards	13	11

This being done, we want to try to figure out the letter-node correspondence, and which letters are missing. Looking at nodes feeding back to themself is also helpful here, as is looking at nodes that are used multiple times in the same list, nodes that are part of a lot of lists, and nodes that have one more arrow going out than in (meaning they're the first item) or the reverse (meaning they're the last). When one list is ambiguous on its own (often because it's not clear which item is missing), looking at other lists usually helps. We do have to be careful about things such as whether ti wraps back to do in musical notes (it doesn't in this puzzle), whether ace is the lowest or highest playing card (lowest), and the use of orthoclase versus fluorite in the Mohs scale (we use orthoclase), though. Eventually, we find that the letters corresponding to each node are as shown here:

It's not really clear what to do with this: the letters mostly seem randomly placed. However, we haven't yet used the missing item from each category (which we probably figured out while getting the letter-node correspondence). Here's each category, with the missing item surrounded in brackets.

Reading the first letters of the missing items, we get PARROT, the puzzle answer.

Authors' Notes

This puzzle concept was a backup puzzle from last year that we ended up not needing to use. (It was also very similar to other sets-of-things puzzles last year.) After trying a tweak this year where all the sets were cyclical, we went with basically the original concept.

Originally, there weren't checkboxes to toggle certain colors (but the emojis were given in correct order). It turned out that most of the difficulty of the puzzle arose from data collection being hard and error-prone, as was revealed from a very short testsolve once the checkboxes were added. Fortunately, reordering the emojis brought this up to a similar level of difficulty as other puzzles in the round. (Each color is still sort of messy, but I tried seeing if it was possible to fully disentangle all the colors and it seemed that it wasn't, so I decided to just go with this because it's at least readable one color at a time.)

I intended the fact that playing cards have three nodes feeding back to themselves, which are adjacent in the path, to be a good breakin. Unfortunately, at least one team found that BOOKKEEPINGS, a very distinctive word known for its three consecutive double letters (though more often in the singular), has the same letter pattern as the playing cards without ten (ATTFFSSENJQK). Sorry to teams that got stuck on this.