Timely Journey

This may have been the hardest Jane Street puzzle I've ever attempted. There was no problem statement. Just the image and the linked PDF below:

So where to even begin? On my first day attempting this puzzle, I analyzed everything I could at face value. We had:

1. A d48 die and a d60 die. The top faces together read 2026.
2. Some Scrabble pieces. Specifically, C$_3$, J$_8$, S$_1$, and I$_1$.
3. Some dominoes. Specifically, 2-0, 1-4, and 0-4.
4. Some suspiciously arranged red and blue pushpins.
5. A globe.
6. A potted plant whose pot reads "Minnesota State High School Mathematics League, 1998, Twin Cities Private Division"
7. The Z pentomino.
8. Eight different colored dice arranged with the number 1 facing up.
9. Some wooden drawer that reads "Hello...."
10. A desktop keyboard.
11. A paper with a crossword-like grid, along with some numbers. Some of the numbers are colored red or blue, and some of the cells are colored red or blue as well.

I figured the objects surrounding the paper had meaning, otherwise the entire puzzle would just be the paper. The answer would most likely lie in the grid on the paper I thought, so I focused on that next.

The first thing I tried to do was make sense of the numbers. It just so happened that there were 12 rows of numbers in the bottom left of the paper, and there were 12 columns of numbers in the top right of the paper. It also looked like there was one red digit and one blue digit for each of the rows and columns of numbers, but one column of numbers had two blue digits, which was strange. Similarly, each row and column of cells in the grid had exactly one red cell and one blue cell. However, these numbers did not end up fitting into the grid, as there were grid entries of length 4 and 13, yet all numbers had length 5, 6, or 7.

Looking at the numbers more closely, I realized that the last two digits in each of the numbers were never $\geq$ 60. Likewise, the 3rd and 4th last digits were never $\geq$ 60 either. So maybe these numbers were actually times! It would have been fitting with the title of the puzzle "Timely Journey". With the "journey" theme and the globe in the image, I figured these numbers were latitudes or longitudes in DDD° MM' SS'' format, where the degrees had 1-3 digits so every 5-7 digit number could be valid. A clue that supported this was that all the degrees in the length 7 numbers were $\leq$ 180, which was the maximum degree for longitudes. It was also suspicious that all the length 7 numbers were vertical, so maybe ALL the vertical numbers were longitudes, as longitudes run north-south. Likewise, all the horizontal numbers had degrees $\leq$ 90, the maximum degree for latitudes, so maybe the horizontal numbers were latitudes.

At this point I had settled on the idea that the horizontal numbers were latitudes and the vertical numbers were longitudes. It was most likely the case that the latitudes were clues for the horizontal entries in the grid, and the longitudes were clues for the vertical entries in the grid. But how to convert these DMS coordinates into things that could fit in the grid? And how did the colored digits come into play? Did they indicate which part of the DMS coordinate to modify somehow?

So I turned back to the objects in the image for more clues. I realized that if I read them the right way, the Scrabble pieces spelled out "C JS 20 16 - 04", which is "see Jane Street 2016-04". Looking at the monthly puzzle of April 2016, there was a puzzle called "Long. Journey" where its given numbers were also in DMS format. So I had at least gotten the format of the numbers correct. Another thing that seemed to be intentionally placed in the image were the dice face up with 1s. With the Z pentomino to their left, it read "Z 1s" or "Z ones". Maybe "Zones"? As in time zones? That made sense with the name of the puzzle and the latitudes/longitudes. Well, it turned out that the plant to the left of the Z pentomino was actually a thyme plant, so it definitely read "Thyme Zones" (time zones)!

So time zones were involved somehow. In total, we had 12 latitudes and 12 longitudes, which is suspiciously the same as the number of UTC offsets. Latitudes and longitudes don't directly correspond to a specific time zone, as latitudes always cross over multiple time zones and longitudes can cross over multiple time zones that are offset by different amounts (e.g. China is all in UTC+8 despite spanning multiple longitudes). However, each longitude does have a unique UTC offset, so I decided to focus on the longitudes first.

From left to right, the longitudes just so happened to fall in each odd UTC offset (+1, +3, ..., +11, -11, ..., -3, -1) if the correct cardinal direction (E or W) was assigned to each longitude. The first six longitudes being East and the last six longitudes being West. This was almost certainly not a coincidence. So if the longitudes got the odd UTC offsets, then surely the latitudes would get the even UTC offsets.

Since the puzzle was called "Timely Journey", we most likely needed to journey around the world through each of the 24 UTC offsets. Then each UTC offset would have a latitude or longitude clue that would reveal a location to visit, which we could then fit into the grid. This seemed like a very reasonable guess at this point, but I had no idea how to choose a location from a latitude or longitude. And if the colored digits meant modifying the latitude/longitude in some way, how would that work?

In case the colored digits did mean modifying the coordinates, I started by choosing a coordinate with the colored digits in the rightmost position. This way, any modification would affect the least significant part of the coordinate. The only coordinate with colored digits in the seconds position was the latitude 21° 14' 08'' with a blue 0 and a red 8. Assuming the seconds would be modified, the only thing interesting I found with latitude 21° 14' N or 21° 14' S were the Cook Islands. Specifically, the island of Rarotonga at 21° 14' S. This seemed promising, as the Cook Islands are in UTC-10, and 21° 14' 08'' lined up with UTC-10 in the latitude list assuming the even UTC offset pattern mentioned before. Perhaps we needed to travel to islands around the world? We also maybe didn't need to modify the coordinates, as 21° 14' 08'' S still landed on Rarotonga.

The next island I found was on 169° 51' 01'' W, which was Niue. The longitude 134° 59' 04'' W landed on the Gambier Islands. The Galápagos were on 0° 42' 29'' S. The Azores were on 25° 37' 56''. Fernando de Noronha was on 3° 50' 56'' S. Martinique was on 14° 39' 57'' N. Madeira was on 32° 45' 06'' N. Bougainville was on 155° 23' 32'' E. But these were the only islands that seemed like real hits. I couldn't find suitable island for 113° 29' 38'' W, which passed through the Americas. There was some island that was a part of Mexico, but it was uninhabited and there was no airport on it. I mean, to get across the ocean, we needed islands. But maybe our coordinates didn't need to land strictly on islands.

Looking at major cities on 113° 29' 38'' W, I found Edmonton. That seemed alright, as the longitude did run straight through the city. Some other cities like Algiers on 03° 03' 19'' E and Nairobi on 36° 49' 27'' E also seemed to work out. For the length 13 grid entry, 30° 35' 28'' E passed through Yekaterinburg, which fit perfectly with Baku (40° 27' 00'' E) for the intersecting length 4 entry. Baku also had UTC+4 and Yekaterinburg had UTC+5, so the time zones might also be connected in order in the grid as well!

After a lot of searching and verifying, I finally had a full list of locations to visit for each of the 24 UTC offsets that also fit into the grid. Here is the full list of locations I ended up with:

Noronha had to be used in place of Fernando de Noronha to fit the grid. All of these locations can be visualized by pasting this json into geojson.io. The final filled-in grid can be found here.

Now the only thing left was to figure out what the red and blue digits and cells meant. We had letters for each of the colored cells, but how did they relate to the colored digits? There was one thing in the image that I hadn't used yet: the pushpins. It was so strange how they were red and blue, at the top of the pushpin pile, and all facing the same direction. They were definitely intentionally placed like this. Looking in the direction the pins were facing, it seemed like they were pointing the "Shift" key on the keyboard the paper was sitting on. Shifting the letters in the colored cells by the colored digits for each latitude and longitude seemed like an obvious thing to do. One of the simplest things you can do with some letters and numbers is use a Caesar cipher after all.

Starting from UTC+0 and going around in increasing UTC offsets until eventually wrapping around to UTC-1, we get the following strings of red and blue letters:

Let's see what happens when we shift these letters by their corresponding colored digits. The number of red digits matched the number of red letters, but this wasn't the case for the blue ones, as there were 24 blue letters and 25 blue digits. However, if we treated the longitude with the two blue digits 1 and 0 as having a shift of 10, then everything would line up perfectly. Below are the shifts for each of the colors:

Performing a right shift, we get the following decoded strings:

The red letters were nicely decoded to almost the entire alphabet in order, and the blue letters were decoded to the question "What red letters are missing?" The only red letters missing were T and Z, so the final answer to this puzzle was $\boxed{\text{TZ}}$. This was kind of an underwhelming answer for such a long puzzle. "TZ" could mean "Time Zones" or "2026" (if you read the letters as numbers). These interpretations of TZ were in the image from the very beginning, so the answer was hidden in plain sight...

I'm very proud of myself for getting this month's puzzle. I was 1 of 53 people to solve it and the 20th person to solve it overall, which was my best rank on a puzzle so far! Having no problem statement and using an image as the only clue probably made it extremely difficult for AI to solve. This explains why this month's puzzle had the lowest solve rate out of recent puzzles, which get roughly 50 correct submissions a day from what I've observed.