diff --git a/src/core/CMakeLists.txt b/src/core/CMakeLists.txt
index 9678047..0cad65c 100644
--- a/src/core/CMakeLists.txt
+++ b/src/core/CMakeLists.txt
@@ -25,7 +25,7 @@ set(KLOTSKI_CORE_SRC
         group/internal/group.cc
 
         ranges/internal/ranges.cc
-        ranges/internal/head.cc
+        ranges/internal/derive.cc
 )
 
 add_library(klotski_core STATIC ${KLOTSKI_CORE_SRC})
diff --git a/src/core/all_cases/all_cases.h b/src/core/all_cases/all_cases.h
index 249b029..4fa9693 100644
--- a/src/core/all_cases/all_cases.h
+++ b/src/core/all_cases/all_cases.h
@@ -58,7 +58,7 @@ typedef std::function<void(std::function<void()>&&)> Executor;
 
 // ------------------------------------------------------------------------------------- //
 
-constexpr auto BASIC_RANGES_NUM = 7311921;
+constexpr auto BASIC_RANGES_NUM = 7311885;
 
 constexpr std::array ALL_CASES_NUM {
     2942906, 2260392, 2942906, 0,
@@ -124,9 +124,6 @@ private:
     /// Get static singleton variable.
     static RangesUnion& get_cases();
 
-    /// Build all valid ranges of the specified head.
-    static void build_cases(int head, Ranges &release);
-
     KLSK_INSTANCE(AllCases)
 };
 
diff --git a/src/core/all_cases/internal/all_cases.cc b/src/core/all_cases/internal/all_cases.cc
index 3d84acc..1da6950 100644
--- a/src/core/all_cases/internal/all_cases.cc
+++ b/src/core/all_cases/internal/all_cases.cc
@@ -2,11 +2,14 @@
 
 #include "all_cases/all_cases.h"
 
+using klotski::cases::Ranges;
 using klotski::cases::AllCases;
+using klotski::cases::BasicRanges;
+using klotski::cases::ALL_CASES_NUM;
 
-/// Calculate all possible klotski heads.
-consteval static std::array<int, 12> case_heads() {
-    std::array<int, 12> heads = {};
+/// Generate all possible klotski heads.
+consteval static std::array<int, 12> heads() {
+    std::array<int, 12> heads {};
     for (int i = 0, head = 0; head < 15; ++head) {
         if (head % 4 != 3) {
             heads[i++] = head;
@@ -15,10 +18,11 @@ consteval static std::array<int, 12> case_heads() {
     return heads;
 }
 
-void AllCases::build_cases(const int head, Ranges &release) {
+/// Build all valid ranges of the specified head.
+static void build_cases(const int head, Ranges &release) {
     release.clear();
     release.reserve(ALL_CASES_NUM[head]);
-    BasicRanges::instance().fetch().with_head(head, release);
+    BasicRanges::instance().fetch().derive(head, release);
 }
 
 void AllCases::build() {
@@ -36,7 +40,7 @@ void AllCases::build_parallel(Executor &&executor) {
         return; // data is already available
     }
     std::vector<std::future<void>> futures;
-    for (auto head : case_heads()) {
+    for (auto head : heads()) {
         auto promise = std::make_shared<std::promise<void>>();
         futures.emplace_back(promise->get_future());
         executor([head, promise = std::move(promise)]() {
@@ -63,10 +67,10 @@ void AllCases::build_parallel_async(Executor &&executor, Notifier &&callback) {
     }
     auto counter = std::make_shared<std::atomic<int>>(0);
     auto all_done = std::make_shared<Notifier>(std::move(callback));
-    for (auto head : case_heads()) {
+    for (auto head : heads()) {
         executor([this, head, counter, all_done]() {
             build_cases(head, get_cases()[head]);
-            if (counter->fetch_add(1) == case_heads().size() - 1) { // all tasks done
+            if (counter->fetch_add(1) == heads().size() - 1) { // all tasks done
                 available_ = true;
                 building_.unlock(); // release building mutex
                 all_done->operator()(); // trigger callback
diff --git a/src/core/all_cases/internal/basic_ranges.cc b/src/core/all_cases/internal/basic_ranges.cc
index 39996f8..ff718d0 100644
--- a/src/core/all_cases/internal/basic_ranges.cc
+++ b/src/core/all_cases/internal/basic_ranges.cc
@@ -8,14 +8,16 @@ using klotski::cases::Ranges;
 using klotski::cases::BasicRanges;
 
 typedef std::tuple<int, int, int> RangeType;
-typedef std::array<RangeType, 204> RangeTypeUnion;
+typedef std::array<RangeType, 203> RangeTypeUnion;
 
 /// Generate all possible basic-ranges permutations.
 consteval static RangeTypeUnion range_types() {
     RangeTypeUnion data;
     for (int i = 0, n = 0; n <= 7; ++n) { // 1x2 + 2x1 -> 0 ~ 7
         for (int n_2x1 = 0; n_2x1 <= n; ++n_2x1) { // 2x1 -> 0 ~ n
-            // TODO: skip n == 7 && n_2x1 == 7
+            if (n == 7 && n_2x1 == 7) {
+                break;
+            }
             for (int n_1x1 = 0; n_1x1 <= (14 - n * 2); ++n_1x1) { // 1x1 -> 0 ~ (14 - 2n)
                 data[i++] = {n, n_2x1, n_1x1};
             }
@@ -48,7 +50,7 @@ void BasicRanges::build_ranges(Ranges &ranges) {
 
     std::list flags { ranges.begin() };
     for (auto [n, n_2x1, n_1x1] : range_types()) {
-        ranges.spawn_more(n, n_2x1, n_1x1);
+        ranges.spawn(n, n_2x1, n_1x1);
         flags.emplace_back(ranges.end()); // mark ordered interval
     }
 
@@ -61,5 +63,7 @@ void BasicRanges::build_ranges(Ranges &ranges) {
         }
     } while (flags.size() > 2); // merge until only one interval remains
 
-    ranges.reverse(); // flip every 2-bit
+    for (auto &x : ranges) {
+        x = range_reverse(x); // flip every 2-bit
+    }
 }
diff --git a/src/core/benchmark/group.cc b/src/core/benchmark/group.cc
index 4cc48aa..d2c6ded 100644
--- a/src/core/benchmark/group.cc
+++ b/src/core/benchmark/group.cc
@@ -16,7 +16,7 @@ using klotski::cases::AllCases;
 static std::vector<uint64_t> all_common_codes() {
     std::vector<uint64_t> codes;
     for (uint64_t head = 0; head < 16; ++head) {
-        for (const auto range : AllCases::instance().fetch()[head].ranges_) {
+        for (const auto range : AllCases::instance().fetch()[head]) {
             codes.emplace_back(head << 32 | range);
         }
     }
@@ -159,7 +159,7 @@ static void SpawnRanges(benchmark::State &state) {
         kk.reserve(7311921);
 
         for (auto [n, n_2x1, n_1x1] : nums) {
-            kk.spawn_more(n, n_2x1, n_1x1);
+            kk.spawn(n, n_2x1, n_1x1);
         }
     }
 
@@ -172,6 +172,35 @@ static void OriginBasicRanges(benchmark::State &state) {
     }
 }
 
+static void OriginAllCases(benchmark::State &state) {
+
+    klotski::cases::BasicRanges::instance().build();
+
+    for (auto _ : state) {
+        auto &pp = klotski::cases::AllCases::instance();
+        pp.available_ = false;
+        pp.build();
+    }
+
+}
+
+static void RangesDerive(benchmark::State &state) {
+
+    auto &basic_ranges = klotski::cases::BasicRanges::instance().fetch();
+
+    klotski::cases::Ranges results;
+    results.reserve(klotski::cases::ALL_CASES_NUM[5]);
+
+    for (auto _ : state) {
+
+        results.clear();
+        basic_ranges.derive(5, results);
+
+    }
+
+    // std::cout << results.size() << " vs " << klotski::cases::ALL_CASES_NUM[5] << std::endl;
+}
+
 // BENCHMARK(CommonCodeToTypeId)->Arg(8)->Arg(64)->Arg(256);
 // BENCHMARK(RawCodeToTypeId)->Arg(8)->Arg(64)->Arg(256);
 
@@ -181,6 +210,10 @@ static void OriginBasicRanges(benchmark::State &state) {
 
 // BENCHMARK(SpawnRanges)->Unit(benchmark::kMillisecond);
 
-BENCHMARK(OriginBasicRanges)->Unit(benchmark::kMillisecond);
+// BENCHMARK(OriginBasicRanges)->Unit(benchmark::kMillisecond);
+
+// BENCHMARK(OriginAllCases)->Unit(benchmark::kMillisecond);
+
+BENCHMARK(RangesDerive)->Unit(benchmark::kMillisecond);
 
 BENCHMARK_MAIN();
diff --git a/src/core/main.cc b/src/core/main.cc
index 010eeef..82cd3a4 100644
--- a/src/core/main.cc
+++ b/src/core/main.cc
@@ -27,14 +27,6 @@ using klotski::codec::SHORT_CODE_LIMIT;
 int main() {
     const auto start = clock();
 
-    // auto kk = klotski::cases::RangesDemo();
-    //
-    // for (auto x : kk) {
-    //     std::cout << x << std::endl;
-    // }
-
-    // klotski::cases::spawn_ranges(2, 1, 4, 4);
-
     // auto raw_code = RawCode::from_common_code(0x1A9BF0C00)->unwrap();
     // auto ret = klotski::cases::group_extend_from_seed(raw_code);
     //
diff --git a/src/core/ranges/internal/derive.cc b/src/core/ranges/internal/derive.cc
new file mode 100644
index 0000000..6c54ea6
--- /dev/null
+++ b/src/core/ranges/internal/derive.cc
@@ -0,0 +1,48 @@
+#include "utils/utility.h"
+#include "ranges/ranges.h"
+
+using klotski::cases::Ranges;
+
+/// Check whether the combination of head and range is valid.
+static int check_range(const int head, uint32_t range) {
+    uint32_t flags = 0b110011 << head; // fill 2x2 block
+    for (int addr = 0, offset = 1; range; range >>= 2, ++offset) { // traverse every 2-bit
+        const auto num = std::countr_one(flags);
+        addr += num; // next unfilled block
+        flags >>= num;
+        switch (range & 0b11) {
+            case 0b00: // space
+            case 0b11: // 1x1 block
+                flags |= 0b1;
+                continue;
+            case 0b01: // 1x2 block
+                if (flags & 0b10 || addr % 4 == 3) { // invalid case
+                    return offset; // broken offset
+                }
+                flags |= 0b11;
+                continue;
+            case 0b10: // 2x1 block
+                if (flags & 0b10000 || addr > 15) { // invalid case
+                    return offset; // broken offset
+                }
+                flags |= 0b10001;
+        }
+    }
+    return 0; // pass check
+}
+
+void Ranges::derive(const int head, Ranges &output) const {
+    for (uint32_t index = 0; index < size(); ++index) {
+        if (const auto offset = check_range(head, (*this)[index])) { // invalid case
+            uint32_t tmp = 1U << (32 - offset * 2); // distance to next possible range
+            ///         !! <- broken
+            /// ( xx xx xx ) xx xx xx ... [reversed range]
+            ///         +1   00 00 00 ...     (delta)
+            tmp += range_reverse((*this)[index]) & ~(tmp - 1);
+            while (range_reverse((*this)[++index]) < tmp) {} // located next range
+            --index;
+            continue;
+        }
+        output.emplace_back(range_reverse((*this)[index])); // release valid case
+    }
+}
diff --git a/src/core/ranges/internal/head.cc b/src/core/ranges/internal/head.cc
deleted file mode 100644
index 6560faa..0000000
--- a/src/core/ranges/internal/head.cc
+++ /dev/null
@@ -1,57 +0,0 @@
-#include "utils/utility.h"
-#include "ranges/ranges.h"
-
-/// Check whether the combination of head and range is valid.
-static int check_range(const int head, uint32_t range) noexcept {
-    constexpr uint32_t M_1x1 = 0b00000001;
-    constexpr uint32_t M_1x2 = 0b00000011;
-    constexpr uint32_t M_2x1 = 0b00010001;
-    constexpr uint32_t M_2x2 = 0b00110011;
-
-    uint32_t flags = M_2x2 << head; // fill 2x2 block
-    for (int addr = 0, offset = 1; range; range >>= 2, ++offset) { // traverse every 2-bit
-        const auto num = klotski::low_zero_num(~flags);
-        addr += num; // next unfilled block
-        flags >>= num;
-        switch (range & 0b11) {
-            case 0b00: // space
-            case 0b11: // 1x1 block
-                flags |= M_1x1;
-                continue;
-            case 0b10: // 2x1 block
-                if ((flags >> 4) & 0b1 || addr > 15) { // invalid case
-                    return offset; // broken offset
-                }
-                flags |= M_2x1;
-                continue;
-            case 0b01: // 1x2 block
-                if ((flags >> 1) & 0b1 || (addr & 0b11) == 0b11) { // invalid case
-                    return offset; // broken offset
-                }
-                flags |= M_1x2;
-                continue;
-        }
-    }
-    return 0; // pass check
-}
-
-void klotski::cases::Ranges::with_head(const int head, Ranges &release) const {
-    // release.clear();
-    // release.reserve(ALL_CASES_NUM[head]);
-    // auto &basic_ranges = BasicRanges::instance().fetch();
-
-    for (uint32_t index = 0; index < ranges_.size(); ++index) {
-        auto offset = check_range(head, ranges_[index]);
-        if (offset) { // invalid case
-            auto tmp = (uint32_t)0b1 << (32 - offset * 2); // distance to next possible range
-            ///         !! <- broken
-            /// ( xx xx xx ) xx xx xx ... [reversed range]
-            ///         +1   00 00 00 ...     (delta)
-            tmp += klotski::range_reverse(ranges_[index]) & ~(tmp - 1);
-            while (klotski::range_reverse(ranges_[++index]) < tmp); // located next range
-            --index;
-            continue;
-        }
-        release.ranges_.emplace_back(klotski::range_reverse(ranges_[index])); // release valid case
-    }
-}
diff --git a/src/core/ranges/internal/ranges.cc b/src/core/ranges/internal/ranges.cc
index c362b68..b95e4ed 100644
--- a/src/core/ranges/internal/ranges.cc
+++ b/src/core/ranges/internal/ranges.cc
@@ -1,54 +1,40 @@
-#include <list>
 #include <algorithm>
 
-#include "ranges/ranges.h"
 #include "utils/utility.h"
+#include "ranges/ranges.h"
 
-template<int N>
-static void demo(std::vector<uint32_t> &ranges, int n_10, int n_11) {
-
-    constexpr auto num = 16 - N;
-    constexpr auto offset = (16 - num) << 1; // offset of low bits
+using klotski::cases::Ranges;
 
-    int n_00 = 16 - N * 2 - n_11;
+template<int N>
+static void build_ranges(std::vector<uint32_t> &ranges, int n_10, int n_11) {
     int n_01 = N - n_10;
+    int n_00 = 16 - N * 2 - n_11;
 
-    std::array<int, num> series {};
-
-    auto kk = std::fill_n(series.begin() + n_00, n_01, 0b01);
-    auto pp = std::fill_n(kk, n_10, 0b10);
-    std::fill_n(pp, n_11, 0b11);
-
-    // std::vector<uint32_t> ranges;
+    std::array<int, 16 - N> series {};
+    std::fill_n(series.begin() + n_00, n_01, 0b01);
+    std::fill_n(series.begin() + n_00 + n_01, n_10, 0b10);
+    std::fill_n(series.begin() + n_00 + n_01 + n_10, n_11, 0b11);
 
     do {
         uint32_t range = 0;
-        for (const auto x : series) // store every 2-bit
+        for (const auto x : series) { // store every 2-bit
             (range <<= 2) |= x;
-        ranges.emplace_back(range << offset);
+        }
+        ranges.emplace_back(range << (N * 2));
     } while (std::ranges::next_permutation(series).found);
-
-    // return ranges;
 }
 
-// void klotski::cases::spawn_ranges(std::vector<uint32_t> &ranges, int n, int n_2x1, int n_1x1) {
-//
-//
-// }
-
-void klotski::cases::Ranges::spawn_more(int n, int n_2x1, int n_1x1) {
-    // spawn_ranges(ranges_, n, n_2x1, n_1x1);
-
+void Ranges::spawn(const int n, const int n_2x1, const int n_1x1) {
+    KLSK_ASSUME(n >= 0 && n_2x1 >= 0 && n_1x1 >= 0);
+    KLSK_ASSUME(n <= 7 && n_2x1 <= n && n_1x1 + n * 2 <= 14);
     switch (n) {
-    case 0: return demo<0>(ranges_, n_2x1, n_1x1);
-    case 1: return demo<1>(ranges_, n_2x1, n_1x1);
-    case 2: return demo<2>(ranges_, n_2x1, n_1x1);
-    case 3: return demo<3>(ranges_, n_2x1, n_1x1);
-    case 4: return demo<4>(ranges_, n_2x1, n_1x1);
-    case 5: return demo<5>(ranges_, n_2x1, n_1x1);
-    case 6: return demo<6>(ranges_, n_2x1, n_1x1);
-    case 7: return demo<7>(ranges_, n_2x1, n_1x1);
-    default: return;
+        case 0: return build_ranges<0>(*this, n_2x1, n_1x1);
+        case 1: return build_ranges<1>(*this, n_2x1, n_1x1);
+        case 2: return build_ranges<2>(*this, n_2x1, n_1x1);
+        case 3: return build_ranges<3>(*this, n_2x1, n_1x1);
+        case 4: return build_ranges<4>(*this, n_2x1, n_1x1);
+        case 5: return build_ranges<5>(*this, n_2x1, n_1x1);
+        case 6: return build_ranges<6>(*this, n_2x1, n_1x1);
+        case 7: return build_ranges<7>(*this, n_2x1, n_1x1);
     }
-
 }
diff --git a/src/core/ranges/ranges.h b/src/core/ranges/ranges.h
index a3a9d39..1ec3ba2 100644
--- a/src/core/ranges/ranges.h
+++ b/src/core/ranges/ranges.h
@@ -3,75 +3,15 @@
 #include <vector>
 #include <cstdint>
 
-#include "utils/utility.h"
-
 namespace klotski::cases {
 
-// void spawn_ranges(std::vector<uint32_t> &ranges, int n, int n_2x1, int n_1x1);
-
-// std::vector<uint32_t> basic_ranges();
-
-// TODO: should we inherit on `std::vector<uint32_t>` ?
-class Ranges {
+class Ranges : public std::vector<uint32_t> {
 public:
-    void spawn_more(int n, int n_2x1, int n_1x1);
-
-    using iterator = std::vector<uint32_t>::iterator;
-    using size_type = std::vector<uint32_t>::size_type;
-    using const_iterator = std::vector<uint32_t>::const_iterator;
-
-    using value_type = std::vector<uint32_t>::value_type;
-
-    using reference = std::vector<uint32_t>::reference;
-
-    using const_reference = std::vector<uint32_t>::const_reference;
-
-    iterator begin() {
-        return ranges_.begin();
-    }
-
-    iterator end() {
-        return ranges_.end();
-    }
-
-    [[nodiscard]] const_iterator begin() const {
-        return ranges_.begin();
-    }
-
-    [[nodiscard]] const_iterator end() const {
-        return ranges_.end();
-    }
-
-    void clear() {
-        ranges_.clear();
-    }
-
-    [[nodiscard]] size_type size() const {
-        return ranges_.size();
-    }
-
-    void reserve(const size_type cap) {
-        ranges_.reserve(cap);
-    }
-
-    void with_head(int head, Ranges &release) const;
-
-    void reverse() {
-        for (auto &x : ranges_) {
-            x = range_reverse(x); // flip every 2-bit
-        }
-    }
-
-    const_reference operator[](const size_type n) const {
-        return ranges_[n];
-    }
-
-    [[nodiscard]] const value_type* data() const {
-        return ranges_.data();
-    }
+    /// Spawn klotski-ranges that match the specified block numbers.
+    void spawn(int n, int n_2x1, int n_1x1);
 
-// private:
-    std::vector<uint32_t> ranges_ {};
+    /// Derive the legal klotski-ranges with specified head.
+    void derive(int head, Ranges &output) const;
 };
 
 } // namespace klotski::cases
diff --git a/src/core/utils/utility.h b/src/core/utils/utility.h
index edb2cb9..3db8b73 100644
--- a/src/core/utils/utility.h
+++ b/src/core/utils/utility.h
@@ -18,6 +18,8 @@
 
 #define KLSK_INLINE __attribute__((always_inline))
 
+#define KLSK_ASSUME(expr) __builtin_assume(expr)
+
 namespace klotski {
 
 /// Get the number of consecutive `0` in the low bits.
diff --git a/src/core_test/cases/all_cases.cc b/src/core_test/cases/all_cases.cc
index 206237c..da950f8 100644
--- a/src/core_test/cases/all_cases.cc
+++ b/src/core_test/cases/all_cases.cc
@@ -39,7 +39,7 @@ protected:
         EXPECT_EQ(all_cases_num, ALL_CASES_NUM_); // verify all cases global size
 
         for (int head = 0; head < 16; ++head) {
-            EXPECT_EQ(hash::xxh3(all_cases[head].ranges_), ALL_CASES_XXH3[head]); // verify all cases checksum
+            EXPECT_EQ(hash::xxh3(all_cases[head]), ALL_CASES_XXH3[head]); // verify all cases checksum
         }
     }
 };
diff --git a/src/core_test/cases/basic_ranges.cc b/src/core_test/cases/basic_ranges.cc
index 526367c..b85aad9 100644
--- a/src/core_test/cases/basic_ranges.cc
+++ b/src/core_test/cases/basic_ranges.cc
@@ -1,7 +1,7 @@
 #include "hash.h"
 #include "helper.h"
 
-static constexpr uint64_t BASIC_RANGES_XXH3 = 0x82b040060044e336;
+static constexpr uint64_t BASIC_RANGES_XXH3 = 0x2ced674494fe904d;
 
 class BasicRangesTest : public testing::Test, public Concurrent {
 protected:
@@ -24,12 +24,12 @@ protected:
     static void Verify() {
         const auto &basic_ranges = BasicRanges::instance().fetch();
         EXPECT_EQ(basic_ranges.size(), BASIC_RANGES_NUM); // verify basic ranges size
-        EXPECT_EQ(hash::xxh3(basic_ranges.ranges_), BASIC_RANGES_XXH3); // verify basic ranges checksum
+        EXPECT_EQ(hash::xxh3(basic_ranges), BASIC_RANGES_XXH3); // verify basic ranges checksum
     }
 };
 
 TEST_FF(BasicRanges, constant) {
-    EXPECT_EQ(BASIC_RANGES_NUM, 7311921);
+    EXPECT_EQ(BASIC_RANGES_NUM, 7311885);
 }
 
 TEST_FF(BasicRanges, basic_ranges) {