diff --git a/src/klotski/all_cases/all_cases.h b/src/klotski/all_cases/all_cases.h
index 4eca3c1..6e83cc3 100644
--- a/src/klotski/all_cases/all_cases.h
+++ b/src/klotski/all_cases/all_cases.h
@@ -1,5 +1,29 @@
 #pragma once
 
+/// AllCases can generate all valid CommonCodes, a total of 29334498 kinds.
+/// For the generated BasicRanges, use different `2x2` block positions to
+/// check respectively. On the `5x4` chessboard, it has 12 possible positions,
+/// and the numbers are distributed in 0 ~ 15, witch called `head`. For the
+/// convenience of calculation, here use an array of length 16 for storage,
+/// of course, the position of 3/7/11/15 will be empty.
+
+///   00 01 02 03
+///   04 05 06 07    00 01 02
+///   08 09 10 11    04 05 06  <- head of 2x2 block
+///   12 13 14 15    08 09 10
+///   16 17 18 19    12 13 14
+
+/// After checking, each head has a different valid `range`, and they are
+/// stored in different arrays to save memory (otherwise the 64-bits length
+/// must be consumed), and all CommonCodes can be exported by using the
+/// following code, which is also integrated in FFI.
+
+///   for (uint64_t head = 0; head < 16; ++head) {
+///       for (const auto &range : AllCases::fetch()[head]) {
+///           printf("%09lX\n", head << 32 | range);
+///       }
+///   }
+
 #include <mutex>
 #include <vector>
 #include <cstdint>
diff --git a/src/klotski/all_cases/basic_ranges.h b/src/klotski/all_cases/basic_ranges.h
index fc097e8..3b06d6c 100644
--- a/src/klotski/all_cases/basic_ranges.h
+++ b/src/klotski/all_cases/basic_ranges.h
@@ -1,5 +1,31 @@
 #pragma once
 
+/// Based on the requirements of valid klotski, the `2x2` block that must exist
+/// and only one, witch will occupy 4 empty slots, and the remaining 16 slots
+/// will be allocated to space, `1x2`, `2x1` and `1x1`. Then, according to the
+/// rules of CommonCode, they are coded as `00` `01` `10` `11` respectively, and
+/// the remaining positions are filled with `0` and stored as 32-bits variables.
+
+/// As we all know, a space or `1x1` block will occupy 1 slot, `1x2` or `2x1`
+/// block will occupy 2 slots, and together they fill 16 positions, so all
+/// possible combinations can be calculated, this number is 204. Each combination
+/// can produce different permutations. After verification, there are a total of
+/// 7311921 possible permutations. The goal of BasicRanges is to find these
+/// permutations, sort them and store them in a `uint32_t` array.
+
+/// In terms of algorithms, there are two options: the first is to generate
+/// out-of-order data and then quickly arrange them; the second is to generate
+/// ordered data for 204 combinations, and then merge and sort them. After testing,
+/// the former is faster in generation (consuming T time), but it will consume
+/// more time in sorting (about 7T), and the latter will cost about 2T in
+/// generation due to the loss of the tree structure queue. But it can save more
+/// time in sorting, which is about 2T, so the second solution will get the result
+/// faster.
+
+/// Finally, due to the performance considerations of AllCases, the resulting data
+/// will be flipped every two bits, which will not consume too much time (less than
+/// 10% of T), but can almost double the speed of the subsequent `check_range`.
+
 #include <mutex>
 #include <vector>
 #include <cstdint>