This puzzle references the Misaka Sisters from A Certain Magical Index/A Certain Scientific Railgun. The Sisters are clones of Misaka Mikoto, a major character in Index, and the protagonist of Railgun. These clones form the Misaka Network, allowing them to communicate and share information with each other. They were part of an experiment where a character named Accelerator would sequentially kill each of them in combat in the hopes of levelling up after killing 20000 Sisters, who would get progressively stronger by learning from previous battles. The experiment ended when a boy named Kamijou Touma defeated Accelerator in battle.

Except for a few special cases (Dolly, Full Tuning, and Misaka Worst), each Sister has a serial number between 1 and 20001. In the grid, there are 5 Misakas whose serial numbers we can initially determine:

• The first one to meet “Big Sister” (Mikoto) was Misaka 9982, whose death was witnessed by Mikoto.
• The last one to be killed in the experiment was Misaka 10031, who was the first to meet “a certain boy” (Kamijou Touma), and whose dead body was later found by him.
• The first one to be saved by “a certain boy” (Kamijou Touma) was Misaka 10032, who later received a heart-shaped necklace from him.
• The one who met Mikoto in Russia and helped her disable a nuclear missile was Misaka 10777.
• The one named Last Order has serial number 20001.

From these 5 initial points, we can start identifying the serial numbers of more Misakas. The grid below shows the serial numbers we can determine from direct mathematical steps without doing any casework.

Note that 20001 is the highest serial number any Misaka has, so the number of Misakas with a serial number created after Misaka 5960 is 20001-5960 = 14041.

From here, we need to do some casework. The Misaka below Misaka 10031 has a serial number of either 7 or 1433, since those are the only prime factors of 10031. When we extend these cases to the nearby section of the grid, we get:

Since the Misaka in the bottom left has a serial number equal to the average of those to the northeast and east, those adjacent serial numbers can’t be 74 and 47, whose average isn’t an integer. This means the one below Misaka 10031 must be Misaka 1433, and the one in the bottom left must be Misaka 16877 or 17786. But only 17786 can be written as the sum of two squares, 17786 = 131^2+25^2 = 17161 + 625, so she must be Misaka 17786, and the one above her must be Misaka 17161. With this resolved, we can fill in the bottom right section of the grid:

Next, the Misaka below Last Order must have a serial number of 177, 339, or 6667. The one to her east would then be Misaka 267, 510, or 10002. But since this serial number must be multiplicatively invertible mod 17824, it must be relatively prime to 17824. The only possibility is 267, and so the one below Last Order must be Misaka 177. With this resolved, we can now fill in the entire right portion of the grid:

We can then get the following cases for the remaining unidentified Misakas:

The top two sets of cases arise from the fact that the Misaka to the west of Misaka 10777 can have a serial number of 13 or 829. And since 12992 = 2^6*7*29, the only possibilities for the Misakas to the west and southwest that form a geometric sequence of factors are:

• 12992, 2^3*7*29 = 1624, 7*29 = 203
• 12992, 2^4*7*29 = 3248, 2^2*7*29 = 812
• 12992, 2^5*7*29 = 6496, 2^4*7*29 = 3248.

The remaining Misaka, in the center left above, has a serial number equal to the average of the serial numbers of the 5 surrounding Misakas. That means that the sum of these numbers must be divisible by 5. The only way to achieve this is to make the top one 5395 and the bottom one 3248, giving the average (5395+13577+12983+12992+3248)/5 = 9639. With this, we can finally complete the grid, which happens to include some additional Misaka Sisters who appeared in the anime:

For extraction, the fact that each row has 5 Misakas, and the reference to Schrödinger’s Cat in the flavortext, clue that we need to extract a binary number from each row, based on whether those specific Misakas are dead (1-10031, grey) or alive (10032-20001, green). We can then convert those binary numbers to letters:

The extracted cluephrase is KIMI NO NA WA ENG. Kimi no Na wa is a Japanese anime film, whose English title is YOUR NAME.

Author's Notes

As a big fan of the Raildex series, Misaka Mikoto, and the Misaka Sisters in particular, I wanted to write a puzzle involving the Sisters. Since they all have serial numbers, I decided that a math puzzle would be the best way to do this. I was somewhat inspired by Twisty Passages from the 2019 MIT Mystery Hunt, but while there were instances of identifying special types of numbers, this puzzle was more about performing mathematical operations to fill out the grid after being given some initial numbers.

I decided to start with 5 of the most important Misaka Sisters in the story, and put one in each column, in a somewhat symmetric arrangement. To extract the correct cluephrase, I only had to make sure that each serial number was in the correct set (dead or alive), but I gave myself some additional constraints: I wanted to include 6 additional Misaka Sisters who appeared in the anime, and I also wanted to have some Misakas between 10000 and 10031 so that extracting binary numbers using a cutoff of 10000 wouldn’t work. Finally, I wanted to have a variety of mathematical operations and special sets of numbers. It took several iterations, but I’m very happy with the result.

The title, Radio Noise , is the name of the project that produced the Sisters, as well as the name of their esper ability. For the flavortext, it was very fortunate that Misaka 10032 wanted to name her black cat Schrödinger, and that Schrödinger’s Cat clued the binary extraction method so well. She also wanted to name it Dog and Tokugawa Ieyasu, but she eventually settled on Later. The fact that about half of the Sisters are dead and about half are alive actually ended up being key to a plot point later on in the Index story, involving the Will of the Whole Misaka Network.

Finally, all Misakas are best girl, but Mikoto is my favorite!