diff --git a/day_21/input.txt b/day_21/input.txt
new file mode 100644
index 0000000..0cad071
--- /dev/null
+++ b/day_21/input.txt
@@ -0,0 +1,5 @@
+540A
+582A
+169A
+593A
+579A
\ No newline at end of file
diff --git a/day_21/solve2.md b/day_21/solve2.md
new file mode 100644
index 0000000..a0b8e84
--- /dev/null
+++ b/day_21/solve2.md
@@ -0,0 +1,78 @@
+
+기본적인 정의를 합니다.
+
+```my-ml-like-sudo-language
+// fg is shortest available pathes function that returns a path from pos to target.
+// type cmd :: enum {
+//   up'^', down'v', right'<', left'>', accept'A'
+// }
+// type path_set = set of list of cmd;
+```
+
+이제 헬퍼 함수를 정이합니다.
+
+```
+// \product is a function that returns a list of all possible combinations of two lists.
+// \product :: path_set -> path_set -> path_set
+// \product [] ys = []
+// \product xs [] = []
+// \product (x:xs) y = (concat x y) ++ (\product xs y)
+// infix 5 \product // associativity is not important. because it commutative.
+
+// ++ is union of two sets.
+// concat :: cmd -> path_set -> path_set
+// concat x [] = [x]
+// concat x (y:ys) = (x:y) ++ (concat x ys)
+
+// product :: path_set -> path_set -> path_set
+// product a b = a \product b
+
+```
+
+```my-ml-like
+
+type start_pos = cmd
+type target_pos = cmd
+type next_pos = cmd
+available_paths ::  start_pos -> target_pos -> path_set
+type Handler 'T :: start_pos -> ('T, next_pos)
+
+// define unit, mappend. these are monoid.
+
+// 
+
+// unit :: Handler of path_set
+// unit = s -> t -> (t, s)
+// mappend :: Handler of path_set -> Handler of path_set -> Handler of path_set
+// mappend f g = s -> 
+//  let (t, s') = f s in
+//  let (a ,s'') = g s'
+//  in ( (product t a )  , s'')
+// 
+// move :: Handler of (cmd -> set of list of cmd)
+
+// move s cmds x = available_paths x \product  
+// accept :: Handler of (list of cmd -> list of cmd)
+
+// f = fg 'A'
+// f :: (list of cmd) -> set of (list of cmd)
+// statement execute 'A' (f (x)) = x
+// f x = move x; accept;
+// f x:tails = (move x; accept; f tails) 'A'
+// sl is get shortest path length
+// sl :: list of cmd -> Int
+// sl cmd = (f cmd) |> (foldr (x,y) -> min(x,y), INF)
+
+// operator \product is a function that returns a list of all possible combinations of two lists.
+// \product :: list of cmd -> list of cmd -> list of cmd
+// \product [] ys = ys
+// \product xs [] = xs
+// \product (x:xs) (y:ys) = x \product (y:ys) ++ xs \product (y:ys)
+// infixr 5 \product
+
+// sl (a \product b) = sl a + sl b
+// f /(.*)A/:rest = f '$1A' \product f rest
+// sl(f /(.*)A/:rest)) = sl (f '$1A' \product f rest)
+//                     = sl (f '$1A') + sl (f rest)
+// 
+```
\ No newline at end of file
diff --git a/day_21/solve_1.test.ts b/day_21/solve_1.test.ts
new file mode 100644
index 0000000..06ae2df
--- /dev/null
+++ b/day_21/solve_1.test.ts
@@ -0,0 +1,270 @@
+import { assertEquals, assertThrows } from "jsr:@std/assert";
+import {
+    Command,
+    DIRPAD_FIRST_POS,
+    executeCommandsOnDirpadRobot,
+    executeCommandsOnNumpadRobot,
+    getDirpadNumber,
+    getNextDirpadRobotState,
+    getNextNumpadRobotState,
+    getNumpadNumber,
+    getPosFromDirpadNumber,
+    getPosFromNumpadNumber,
+    NUMPAD_FIRST_POS,
+    Pos,
+    shortestPathesOnDirpadRobot,
+    shortestPathesOnNumpadRobot,
+    solve_1,
+} from "./solve_1.ts";
+
+Deno.test("getNumbericKeypadNumber", () => {
+    assertEquals(getNumpadNumber([0, 0]), "7");
+    assertEquals(getNumpadNumber([1, 0]), "8");
+    assertEquals(getNumpadNumber([2, 0]), "9");
+    assertEquals(getNumpadNumber([0, 1]), "4");
+    assertEquals(getNumpadNumber([1, 1]), "5");
+    assertEquals(getNumpadNumber([2, 1]), "6");
+    assertEquals(getNumpadNumber([0, 2]), "1");
+    assertEquals(getNumpadNumber([1, 2]), "2");
+    assertEquals(getNumpadNumber([2, 2]), "3");
+    assertEquals(getNumpadNumber([0, 3]), " ");
+    assertEquals(getNumpadNumber([1, 3]), "0");
+    assertEquals(getNumpadNumber([2, 3]), "A");
+});
+
+Deno.test("getPosFromNumpadNumber", () => {
+    assertEquals(getPosFromNumpadNumber("7"), [0, 0]);
+    assertEquals(getPosFromNumpadNumber("8"), [1, 0]);
+    assertEquals(getPosFromNumpadNumber("9"), [2, 0]);
+    assertEquals(getPosFromNumpadNumber("4"), [0, 1]);
+    assertEquals(getPosFromNumpadNumber("5"), [1, 1]);
+    assertEquals(getPosFromNumpadNumber("6"), [2, 1]);
+    assertEquals(getPosFromNumpadNumber("1"), [0, 2]);
+    assertEquals(getPosFromNumpadNumber("2"), [1, 2]);
+    assertEquals(getPosFromNumpadNumber("3"), [2, 2]);
+    assertEquals(getPosFromNumpadNumber("0"), [1, 3]);
+    assertEquals(getPosFromNumpadNumber("A"), [2, 3]);
+});
+
+Deno.test("getNextKeypadRobotState", () => {
+    assertThrows(() => getNextNumpadRobotState({ pos: [0, 0] }, "^"));
+    assertEquals(getNextNumpadRobotState({ pos: [0, 0] }, "v"), {
+        pos: [0, 1],
+    });
+    assertThrows(() => getNextNumpadRobotState({ pos: [0, 0] }, "<"));
+    assertEquals(getNextNumpadRobotState({ pos: [0, 0] }, ">"), {
+        pos: [1, 0],
+    });
+    assertEquals(getNextNumpadRobotState({ pos: [0, 0] }, "A"), {
+        pos: [0, 0],
+    });
+
+    assertEquals(getNextNumpadRobotState({ pos: [1, 1] }, "^"), {
+        pos: [1, 0],
+    });
+    assertEquals(getNextNumpadRobotState({ pos: [1, 1] }, "v"), {
+        pos: [1, 2],
+    });
+    assertEquals(getNextNumpadRobotState({ pos: [1, 1] }, "<"), {
+        pos: [0, 1],
+    });
+    assertEquals(getNextNumpadRobotState({ pos: [1, 1] }, ">"), {
+        pos: [2, 1],
+    });
+    assertEquals(getNextNumpadRobotState({ pos: [1, 1] }, "A"), {
+        pos: [1, 1],
+    });
+});
+
+Deno.test("getDirKeypadNumber", () => {
+    assertEquals(getDirpadNumber([0, 0]), " ");
+    assertEquals(getDirpadNumber([1, 0]), "^");
+    assertEquals(getDirpadNumber([2, 0]), "A");
+    assertEquals(getDirpadNumber([0, 1]), "<");
+    assertEquals(getDirpadNumber([1, 1]), "v");
+    assertEquals(getDirpadNumber([2, 1]), ">");
+});
+
+Deno.test("getPosFromDirKeypadNumber", () => {
+    assertEquals(getPosFromDirpadNumber(" "), [0, 0]);
+    assertEquals(getPosFromDirpadNumber("^"), [1, 0]);
+    assertEquals(getPosFromDirpadNumber("A"), [2, 0]);
+    assertEquals(getPosFromDirpadNumber("<"), [0, 1]);
+    assertEquals(getPosFromDirpadNumber("v"), [1, 1]);
+    assertEquals(getPosFromDirpadNumber(">"), [2, 1]);
+});
+
+Deno.test("getNextDirKeypadRobotState", () => {
+    assertEquals(getNextDirpadRobotState({ pos: [1, 1] }, "^"), {
+        pos: [1, 0],
+    });
+    assertThrows(() => getNextDirpadRobotState({ pos: [1, 1] }, "v"));
+    assertEquals(getNextDirpadRobotState({ pos: [1, 1] }, "<"), {
+        pos: [0, 1],
+    });
+    assertEquals(getNextDirpadRobotState({ pos: [1, 1] }, ">"), {
+        pos: [2, 1],
+    });
+    assertEquals(getNextDirpadRobotState({ pos: [1, 1] }, "A"), {
+        pos: [1, 1],
+    });
+});
+
+Deno.test("executeCommandsOnNumpadRobot", () => {
+    const commands = [
+        "<A^A>^^AvvvA".split("") as Command[],
+        "<A^A^>^AvvvA".split("") as Command[],
+        "<A^A^^>AvvvA".split("") as Command[],
+    ];
+    for (const command of commands) {
+        const out: string[] = [];
+        executeCommandsOnNumpadRobot(
+            {
+                pos: NUMPAD_FIRST_POS,
+            },
+            command,
+            (o) => {
+                out.push(o);
+            },
+        );
+        assertEquals(out, ["0", "2", "9", "A"]);
+    }
+});
+
+Deno.test("executeCommandsOnDirKeypadRobot", () => {
+    const input_result = [
+        [
+            "<vA<AA>>^AvAA<^A>A<v<A>>^AvA^A<vA>^A<v<A>^A>AAvA^A<v<A>A>^AAAvA<^A>A"
+                .split("") as Command[],
+            "v<<A>>^A<A>AvA<^AA>A<vAAA>^A",
+        ],
+        ["v<<A>>^A<A>AvA<^AA>A<vAAA>^A".split("") as Command[], "<A^A>^^AvvvA"],
+    ] as const;
+    for (const [input, result] of input_result) {
+        const out: string[] = [];
+        executeCommandsOnDirpadRobot(
+            {
+                pos: DIRPAD_FIRST_POS,
+            },
+            input,
+            (o) => {
+                out.push(o);
+            },
+        );
+        assertEquals(out.join(""), result);
+    }
+});
+
+Deno.test("shortestPathesOnNumpadRobot", () => {
+    const testCase: {
+        pos: Pos;
+        target: string;
+        expect: string[][];
+        msg: string;
+    }[] = [
+        {
+            pos: NUMPAD_FIRST_POS,
+            target: "A",
+            expect: [[]],
+            msg: "should return empty array when target is same as pos",
+        },
+        {
+            pos: [0, 0],
+            target: "7",
+            expect: [[]],
+            msg: "should return empty array when target is same as pos",
+        },
+        {
+            pos: NUMPAD_FIRST_POS,
+            target: "9",
+            expect: [["^", "^", "^"]],
+            msg: "should return shortest path to target",
+        },
+        {
+            pos: [0, 0],
+            target: "5",
+            expect: [["v", ">"], [">", "v"]],
+            msg: "should return shortest path to target",
+        },
+        {
+            pos: [1, 3],
+            target: "1",
+            expect: [["^", "<"]],
+            msg: "should return shortest path to target",
+        },
+    ];
+    for (const { pos, target, expect, msg } of testCase) {
+        assertEquals(shortestPathesOnNumpadRobot({ pos }, target), expect, msg);
+    }
+});
+
+Deno.test("shortestPathesOnDirpadRobot", () => {
+    const testCase: {
+        pos: Pos;
+        target: string;
+        expect: string[][];
+        msg: string;
+    }[] = [
+        {
+            pos: DIRPAD_FIRST_POS,
+            target: "A",
+            expect: [["A"]],
+            msg: "should return empty array when target is same as pos",
+        },
+        {
+            pos: [1, 0],
+            target: "^",
+            expect: [["A"]],
+            msg: "should return empty array when target is same as pos",
+        },
+        {
+            pos: DIRPAD_FIRST_POS,
+            target: ">",
+            expect: [["v", "A"]],
+            msg: "should return shortest path to target",
+        },
+        {
+            pos: [1, 0],
+            target: ">",
+            expect: [["v", ">", "A"], [">", "v", "A"]],
+            msg: "should return shortest path to target",
+        },
+        {
+            pos: [0, 1],
+            target: "^",
+            expect: [[">", "^", "A"]],
+            msg: "should return shortest path to target",
+        },
+    ];
+    for (const { pos, target, expect, msg } of testCase) {
+        assertEquals(shortestPathesOnDirpadRobot({ pos }, target), expect, msg);
+    }
+});
+
+Deno.test("example test", () => {
+    const testCase: [string, string][] = [
+        [
+            "029A",
+            "<vA<AA>>^AvAA<^A>A<v<A>>^AvA^A<vA>^A<v<A>^A>AAvA^A<v<A>A>^AAAvA<^A>A",
+        ],
+        [
+            "980A",
+            "<v<A>>^AAAvA^A<vA<AA>>^AvAA<^A>A<v<A>A>^AAAvA<^A>A<vA>^A<A>A",
+        ],
+        [
+            "179A",
+            "<v<A>>^A<vA<A>>^AAvAA<^A>A<v<A>>^AAvA^A<vA>^AA<A>A<v<A>A>^AAAvA<^A>A",
+        ],
+        [
+            "456A",
+            "<v<A>>^AA<vA<A>>^AAvAA<^A>A<vA>^A<A>A<vA>^A<A>A<v<A>A>^AAvA<^A>A",
+        ],
+        [
+            "379A",
+            "<v<A>>^AvA^A<vA<AA>>^AAvA<^A>AAvA^A<vA>^AA<A>A<v<A>A>^AAAvA<^A>A",
+        ],
+    ];
+    for (const [input, expect] of testCase) {
+        assertEquals(solve_1(input), expect.length);
+    }
+});
diff --git a/day_21/solve_1.ts b/day_21/solve_1.ts
new file mode 100644
index 0000000..65dc51f
--- /dev/null
+++ b/day_21/solve_1.ts
@@ -0,0 +1,254 @@
+export type Pos = [number, number];
+export type NumpadRobotState = {
+    pos: Pos;
+};
+export type Command = "^" | "v" | "<" | ">" | "A";
+
+export function getNextPos(pos: Pos, command: Command): Pos {
+    const [x, y] = pos;
+    switch (command) {
+        case "^":
+            return [x, y - 1];
+        case "v":
+            return [x, y + 1];
+        case "<":
+            return [x - 1, y];
+        case ">":
+            return [x + 1, y];
+        default:
+            return pos;
+    }
+}
+
+const NumbericKeypad = "789\n456\n123\n 0A";
+export const NUMPAD_FIRST_POS: Pos = [2, 3];
+export function getNumpadNumber([x, y]: Pos): string {
+    return NumbericKeypad[y * 4 + x];
+}
+export function getPosFromNumpadNumber(num: string): Pos {
+    const idx = NumbericKeypad.indexOf(num);
+    return [idx % 4, Math.floor(idx / 4)];
+}
+
+export function getNextNumpadRobotState(
+    state: NumpadRobotState,
+    command: Command,
+): NumpadRobotState {
+    const nextPos = getNextPos(state.pos, command);
+    const [nx, ny] = nextPos;
+    if (
+        nx < 0 || nx > 2 || ny < 0 || ny > 3 ||
+        getNumpadNumber(nextPos) === " "
+    ) {
+        throw new Error(
+            `Invalid command: ${command} for pos: ${state.pos.join(",")} -> ${
+                nextPos.join(",")
+            } is out of bound`,
+        );
+    }
+    return { pos: nextPos };
+}
+
+const DirectionalKeypad = " ^A\n<v>";
+export const DIRPAD_FIRST_POS: Pos = [2, 0];
+export type DirKeypadRobotState = {
+    pos: Pos;
+};
+export function getDirpadNumber([x, y]: Pos): string {
+    return DirectionalKeypad[y * 4 + x];
+}
+export function getPosFromDirpadNumber(num: string): Pos {
+    const idx = DirectionalKeypad.indexOf(num);
+    return [idx % 4, Math.floor(idx / 4)];
+}
+
+export function getNextDirpadRobotState(
+    state: DirKeypadRobotState,
+    command: Command,
+): DirKeypadRobotState {
+    const nextPos = getNextPos(state.pos, command);
+    const [nx, ny] = nextPos;
+    if (
+        nx < 0 || nx > 2 || ny < 0 || ny > 1 ||
+        getDirpadNumber(nextPos) === " "
+    ) {
+        throw new Error(
+            `Invalid command: ${command} for pos: ${state.pos.join(",")} -> ${
+                nextPos.join(",")
+            } is out of bound`,
+        );
+    }
+    return { pos: nextPos };
+}
+
+export function executeCommandsOnNumpadRobot(
+    state: NumpadRobotState,
+    commands: Command[],
+    output: (out: string) => void,
+): NumpadRobotState {
+    return commands.reduce((state, command) => {
+        if (command === "A") {
+            output(getNumpadNumber(state.pos));
+        }
+        return getNextNumpadRobotState(state, command);
+    }, state);
+}
+
+export function executeCommandsOnDirpadRobot(
+    state: DirKeypadRobotState,
+    commands: Command[],
+    output: (out: string) => void,
+): DirKeypadRobotState {
+    return commands.reduce((state, command) => {
+        if (command === "A") {
+            output(getDirpadNumber(state.pos));
+        }
+        return getNextDirpadRobotState(state, command);
+    }, state);
+}
+
+export function shortestPathesOnNumpadRobot(
+    state: NumpadRobotState,
+    targetNum: string,
+): Command[][] {
+    const targetPos = getPosFromNumpadNumber(targetNum);
+    const len = Math.abs(targetPos[0] - state.pos[0]) +
+        Math.abs(targetPos[1] - state.pos[1]);
+    const queue: [Command[], NumpadRobotState][] = [[[], state]];
+    const paths: Command[][] = [];
+    while (queue.length > 0) {
+        const [path, state] = queue.shift()!;
+        if (path.length === len) {
+            if (
+                state.pos[0] === targetPos[0] && state.pos[1] === targetPos[1]
+            ) {
+                paths.push(path);
+            }
+            continue;
+        }
+        for (const command of ["^", "v", "<", ">"] as Command[]) {
+            try {
+                const nextState = getNextNumpadRobotState(state, command);
+                queue.push([path.concat(command), nextState]);
+            } catch (e) {
+                // Ignore invalid command
+            }
+        }
+    }
+    return paths;
+}
+
+export function shortestPathesOnDirpadRobot(
+    state: DirKeypadRobotState,
+    targetNum: string,
+): Command[][] {
+    const targetPos = getPosFromDirpadNumber(targetNum);
+    const len = Math.abs(targetPos[0] - state.pos[0]) +
+        Math.abs(targetPos[1] - state.pos[1]);
+    const queue: [Command[], DirKeypadRobotState][] = [[[], state]];
+    const paths: Command[][] = [];
+    while (queue.length > 0) {
+        const [path, state] = queue.shift()!;
+        if (path.length === len) {
+            if (
+                state.pos[0] === targetPos[0] && state.pos[1] === targetPos[1]
+            ) {
+                paths.push(path);
+            }
+            continue;
+        }
+        for (const command of ["^", "v", "<", ">"] as Command[]) {
+            try {
+                const nextState = getNextDirpadRobotState(state, command);
+                queue.push([path.concat(command), nextState]);
+            } catch (e) {
+                // Ignore invalid command
+            }
+        }
+    }
+    return paths.map((p) => p.concat("A"));
+}
+
+export function StepShortestPathesOnDirpadRobot(
+    state: DirKeypadRobotState,
+    targetNums: string[],
+) {
+    return targetNums.reduce( (acc, targetNum) => {
+        const prevPos = acc.prevPos;
+        const paths = shortestPathesOnDirpadRobot({ pos: prevPos }, targetNum);
+        if (paths.length === 0) {
+            throw new Error(`No path found for ${targetNum} from ${prevPos}`);
+        }
+        return {
+            prevPos: getPosFromDirpadNumber(targetNum),
+            // product of all
+            paths: acc.paths.flatMap((p) => paths.map((np) => p.concat(np))),
+        }
+    }, {
+        prevPos: state.pos,
+        paths: [[]] as Command[][],
+    }).paths;
+}
+
+function filterShortestPathes(paths: Command[][]) {
+    let len = Number.MAX_SAFE_INTEGER;
+    for (const path of paths) {
+        len = Math.min(len, path.length);
+    }
+    // len = Math.min(...paths.map((p) => p.length)); // this is call stack overflow...
+    // because the array is too big, the argument list is too long.
+    // so we need to calculate the min value by ourself.
+    return paths.filter((p) => p.length === len);
+}
+
+export function solve_1(code: string) {
+    const numbers = code.split("");
+    let start = NUMPAD_FIRST_POS;
+
+    const finalOuts: string[] = [];
+
+    for (const num of numbers) {
+        console.log("num", num);
+        const numpadPaths = shortestPathesOnNumpadRobot({ pos: start }, num).map((p) => p.concat("A"));
+        if (numpadPaths.length === 0) {
+            throw new Error(`No path found for ${num} from ${start}`);
+        }
+        console.log("avail path", numpadPaths.map((p) => p.join("")));
+        const pathes1 = filterShortestPathes(numpadPaths.flatMap((numpadPath) => {
+            return StepShortestPathesOnDirpadRobot({ pos: DIRPAD_FIRST_POS }, numpadPath);
+        }));
+
+        console.log("path1",pathes1.map((p) => p.join("")));
+        const pathes2 = filterShortestPathes(pathes1.flatMap((path) => {
+            return StepShortestPathesOnDirpadRobot({ pos: DIRPAD_FIRST_POS }, path);
+        }));
+        console.log("path2",pathes2.map((p) => p.join("")));
+        const pathes3 = filterShortestPathes(pathes2.flatMap((path) => {
+            return StepShortestPathesOnDirpadRobot({ pos: DIRPAD_FIRST_POS }, path);
+        }));
+        // console.log("path3",pathes3.map((p) => p.join("")));
+        const executePath3 = (path: Command[]) => {
+            const outs: string[] = [];
+            executeCommandsOnDirpadRobot({ pos: DIRPAD_FIRST_POS }, path, (o) => { outs.push(o); });
+            console.log("outs", outs.join(""));
+            return outs;
+        };
+        finalOuts.push(...executePath3(pathes3[0]));
+        start = getPosFromNumpadNumber(num);
+    }
+    console.log("finalOuts", finalOuts.join(""), finalOuts.length);
+    return finalOuts.length;
+}
+
+
+if (import.meta.main) {
+    const codes = Deno.readTextFileSync("input.txt").replaceAll("\r","").split("\n");
+    let total = 0;
+    for (const code of codes){
+        const len = solve_1(code);
+        const n = parseInt(code.slice(0,3));
+        console.log("len", len,  n, len * n);
+        total += n * len;
+    }
+    console.log("total", total);
+}
\ No newline at end of file
diff --git a/day_21/solve_2.test.ts b/day_21/solve_2.test.ts
new file mode 100644
index 0000000..4831f0d
--- /dev/null
+++ b/day_21/solve_2.test.ts
@@ -0,0 +1,124 @@
+import { assertEquals } from "jsr:@std/assert/equals";
+import {
+    availableShortestPaths,
+    Command,
+    inputCommand,
+    inputCommands,
+    Path,
+    PathSetHandler,
+    shortestPathLength,
+} from "./solve_2.ts";
+
+Deno.test("availableShortestPaths - blocked", () => {
+    const paths = availableShortestPaths("<", "^");
+    assertEquals(paths.map((x) => x.toString()), [">^"]);
+});
+
+Deno.test("availableShortestPaths - common", () => {
+    const paths = availableShortestPaths(">", "^");
+    assertEquals(paths.map((x) => x.toString()).sort(), ["<^", "^<"].sort());
+});
+
+Deno.test("Path - length", () => {
+    const path = new Path(["^", "v", "<", ">", "A"]);
+    assertEquals(path.length, 5);
+});
+
+Deno.test("Path - execute", () => {
+    const path = new Path(["<", "v", "A", ">", "A"]);
+    const ret: Command[] = [];
+    path.execute("A", (c) => {
+        ret.push(c);
+    });
+    assertEquals(ret.join(""), "v>");
+});
+
+Deno.test("Path - concat", () => {
+    const path = new Path(["<", "v", "A", ">", "A"]);
+    assertEquals(path.concat("^").toString(), "<vA>A^");
+});
+
+Deno.test("PathHandler - coproduct", () => {
+    const lhs = PathSetHandler.fromPathSet([Path.from(["A"])]);
+    const rhs = PathSetHandler.fromPathSet([Path.from(["v"])]);
+    const set = lhs.coproduct(rhs);
+    const [m, _] = set.call(">");
+    assertEquals(
+        m.map((x) => x.toString()).sort(),
+        [
+            "A",
+            "v",
+        ].sort(),
+    );
+});
+Deno.test("PathHandler - product", () => {
+    const lhs = PathSetHandler.fromPathSet([Path.from(["A"])]);
+    const rhs = PathSetHandler.fromPathSet([Path.from(["v"])]);
+    const set = lhs.product(rhs);
+    const [m, _] = set.call(">");
+    assertEquals(
+        m.map((x) => x.toString()).sort(),
+        [
+            "Av",
+        ].sort(),
+    );
+});
+
+Deno.test("inputCommand", () => {
+    const handler = inputCommand("^");
+    const [m, a] = handler.call(">");
+    assertEquals(
+        m.map((x) => x.toString()).sort(),
+        [
+            "<^A",
+            "^<A",
+        ].sort(),
+    );
+});
+
+Deno.test("inputCommands", () => {
+    const handler = inputCommands(Path.from([">", "A"]));
+    const [m, a] = handler.call(">");
+    assertEquals(
+        m.map((x) => x.toString()).sort(),
+        [
+            "A^A",
+        ].sort(),
+    );
+});
+
+Deno.test("shortestPathLength", () => {
+    const inputcmdsShort = (start: Command,
+        cmds: Command[]) => shortestPathLength(inputCommands(Path.from(cmds)))(start)[0];
+    
+    assertEquals(inputcmdsShort("A", (["A"])), 1);
+    assertEquals(inputcmdsShort("A", (["A", ">"])), 3);
+    assertEquals(inputcmdsShort("A", (["A", ">", "^"])), 6);
+
+    assertEquals(inputcmdsShort("A", (["<"])), 4);
+    assertEquals(inputcmdsShort("A", (["<", "A"])), 8);
+});
+
+Deno.test("shortestPathLength - property", () => {
+    // random path list
+    const pathes = [
+        Path.from(["^", "v", "<", ">", "A"]),
+        Path.from(["^", "v", "<", ">", "A", "v"]),
+        Path.from(["^", "v", "<", ">", "A", "v", "<"]),
+        Path.from(["^", "v", "<", ">", "A", "v", "<", ">"]),
+        Path.from(["A", "v", "v", ">", ">", "v", ">", "v"]),
+    ];
+    for (const paths of pathes) {
+        for (const s of ["^", "v", "<", ">", "A"] as Command[]) {
+            const [x, ...xs] = paths;
+            const s1 = shortestPathLength(inputCommands(Path.from([x])))(s)[0];
+            const s2 = shortestPathLength(inputCommands(Path.from(xs)))(x)[0];
+            const s3 = shortestPathLength(inputCommands(paths))(s)[0];
+            assertEquals(
+                s1 + s2,
+                s3,
+                `s: ${s}, x: ${x}, xs: ${xs}, s1: ${s1}, s2: ${s2}, s3: ${s3}`,
+            );
+        }
+    }
+});
diff --git a/day_21/solve_2.ts b/day_21/solve_2.ts
new file mode 100644
index 0000000..3daac19
--- /dev/null
+++ b/day_21/solve_2.ts
@@ -0,0 +1,460 @@
+// fg is shortest available pathes function that returns a path from pos to target.
+// type cmd :: enum {
+//   up'^', down'v', right'<', left'>', accept'A'
+
+import { getPosFromNumpadNumber, NUMPAD_FIRST_POS, shortestPathesOnNumpadRobot } from "./solve_1.ts";
+
+// }
+export type Command = "^" | "v" | "<" | ">" | "A";
+export type Pos = [number, number];
+// type path = list of cmd;
+// type path_set = set of path;
+export type PathSet = Path[];
+
+// |--|--|--|
+// |  |^ | A|
+// |--|--|--|
+// |< |v | >|
+// |--|--|--|
+const lookupCommandPos: Record<Command, Pos> = {
+    "^": [1, 0],
+    "v": [1, 1],
+    "<": [0, 1],
+    ">": [2, 1],
+    "A": [2, 0],
+};
+export function commandToPos(command: Command): Pos {
+    return lookupCommandPos[command];
+}
+export function posToCommand(pos: Pos): Command {
+    for (const [key, value] of Object.entries(lookupCommandPos)) {
+        if (value[0] === pos[0] && value[1] === pos[1]) {
+            return key as Command;
+        }
+    }
+    throw new Error(`invalid pos: ${pos}`);
+}
+
+export class Path {
+    value: string;
+    constructor(commands: Command[]) {
+        this.value = commands.join("");
+    }
+    get length() {
+        return this.value.length;
+    }
+    [Symbol.iterator](): IterableIterator<Command> {
+        return this.value[Symbol.iterator]() as IterableIterator<Command>;
+    }
+    first() {
+        return this.value[0] as Command;
+    }
+    rest() {
+        const ret = new Path([]);
+        ret.value = this.value.slice(1);
+        return ret;
+    }
+    map<T>(fn: (cmd: Command) => T) {
+        const result: T[] = [];
+        for (const cmd of this.value) {
+            result.push(fn(cmd as Command));
+        }
+        return result;
+    }
+    flatMap<T>(fn: (cmd: Command) => T[]) {
+        const result: T[] = [];
+        for (const cmd of this.value) {
+            result.push(...fn(cmd as Command));
+        }
+    }
+    reduce<T>(fn: (acc: T, cmd: Command) => T, init: T) {
+        let acc = init;
+        for (const cmd of this.value) {
+            acc = fn(acc, cmd as Command);
+        }
+        return acc;
+    }
+    get last() {
+        return this.value[this.value.length - 1];
+    }
+    toString() {
+        return this.value;
+    }
+    toCommands() {
+        return this.value.split("") as Command[];
+    }
+    // ++ is union of two sets.
+    // concat :: cmd -> path_set -> path_set
+    // concat x [] = [x]
+    // concat x (y:ys) = (y:x) ++ (concat x ys)
+    concat(command: Command) {
+        const ret = new Path([]);
+        ret.value = this.value + command;
+        return ret;
+    }
+    concatPath(path: Path) {
+        const ret = new Path([]);
+        ret.value = this.value + path.value;
+        return ret;
+    }
+    static from(commands: Command[]) {
+        return new Path(commands);
+    }
+    // \product is a function that returns a list of all possible combinations of two lists.
+    // \product :: path_set -> path_set -> path_set
+    // \product x [] = []
+    // \product (x:xs) y = (concat x y) ++ (\product xs y)
+    // infix 5 \product // associativity is not important. because it commutative.
+    // product :: path_set -> path_set -> path_set
+    // product a b = a \product b
+
+    static product(a: PathSet, b: PathSet): PathSet {
+        return a.flatMap((x) => b.map((y) => x.concatPath(y)));
+    }
+
+    // it's not pure function.
+    // only use for test.
+    // execute :: (cmd, cmd -> never) -> Option of next_pos
+    execute(start: Command, onAccept: (accept: Command) => void): Pos | null {
+        let pos = commandToPos(start);
+        for (const command of this.toCommands()) {
+            if (command === "A") {
+                onAccept(posToCommand(pos));
+                continue;
+            }
+            const newPos = moveSingle(pos, command);
+            if (newPos === null) {
+                return null;
+            }
+            pos = newPos;
+        }
+        return pos;
+    }
+
+    equals(other: Path) {
+        return this.value === other.value;
+    }
+}
+
+const lookupCommandDelta: Record<Command, Pos> = {
+    "^": [0, -1],
+    "v": [0, 1],
+    "<": [-1, 0],
+    ">": [1, 0],
+    "A": [0, 0],
+};
+export function pathToDelta(command: Command): Pos {
+    return lookupCommandDelta[command];
+}
+export function moveSingle(pos: Pos, command: Command): Pos | null {
+    const [x, y] = pos;
+    const [dx, dy] = pathToDelta(command);
+    const newPos: [number, number] = [x + dx, y + dy];
+    // out of range
+    if (newPos[0] < 0 || newPos[0] > 2 || newPos[1] < 0 || newPos[1] > 1) {
+        return null;
+    }
+    // invalid position
+    if (newPos[0] === 0 && newPos[1] === 0) {
+        return null;
+    }
+    return newPos;
+}
+
+// type Handler 'T :: start_pos -> ('T, next_pos)
+export type Handler<T> = (start: Command) => [T, Command];
+export class PathSetHandler {
+    private fn: Handler<PathSet>;
+    constructor(fn: Handler<PathSet>) {
+        this.fn = fn;
+    }
+    // call :: Handler of start_pos -> (path_set, next_pos)
+    // call f = f
+    call(command: Command) {
+        return this.fn(command);
+    }
+
+    // mempty :: Handler of path_set
+    // mempty = \s -> ([], s)
+    static empty() {
+        return new PathSetHandler(
+            (s: Command) => [[], s],
+        );
+    }
+    // munit :: Handler of path_set
+    // munit = \s -> ([[]], s)
+    static unit() {
+        return new PathSetHandler(
+            (s: Command) => [[Path.from([])], s],
+        );
+    }
+    static fromPathSet(t: PathSet) {
+        return new PathSetHandler(
+            (s: Command) => [t, s],
+        );
+    }
+
+    coproduct(f: PathSetHandler) {
+        // union of two sets.
+        return new PathSetHandler(
+            (s: Command) => {
+                const [t, s1] = this.fn(s);
+                const [a, s2] = f.call(s1);
+                // merge two sets t and a.
+                const set = new Set([...t, ...a]);
+                const result = Array.from(set);
+                return [result, s2];
+            },
+        );
+    }
+    // mproduct :: Handler of path_set -> Handler of path_set -> Handler of path_set
+    // mproduct f g = s ->
+    //  let (t, s') = f s in
+    //  let (a ,s'') = g s'
+    //  in ( (product t a )  , s'')
+    // op ; = mproduct
+    // infixr 5 ;
+    product(f: PathSetHandler) {
+        return new PathSetHandler(
+            (s: Command) => {
+                const [t, s1] = this.fn(s);
+                const [a, s2] = f.fn(s1);
+                return [Path.product(t, a), s2];
+            },
+        );
+    }
+
+    // available_paths ::  (start_pos, target_pos) -> path_set
+}
+
+// type start_pos = cmd
+// type target_pos = cmd
+// type next_pos = cmd
+// available_paths ::  (start_pos, target_pos) -> path_set
+export function availableShortestPaths(
+    start: Command,
+    target: Command,
+): PathSet {
+    const startPos = commandToPos(start);
+    const targetPos = commandToPos(target);
+    const paths: Path[] = [];
+    const queue: [Path, Pos][] = [[new Path([]), startPos]];
+    const deltaSum = Math.abs(targetPos[0] - startPos[0]) +
+        Math.abs(targetPos[1] - startPos[1]);
+    while (queue.length > 0) {
+        const [path, pos] = queue.shift()!;
+        if (pos[0] === targetPos[0] && pos[1] === targetPos[1]) {
+            paths.push(path);
+            continue;
+        }
+        if (path.length >= deltaSum) {
+            continue;
+        }
+        for (const command of ["^", "v", "<", ">"] as Command[]) {
+            const nextPos = moveSingle(pos, command);
+            if (nextPos) {
+                queue.push([path.concat(command), nextPos]);
+            }
+        }
+    }
+    if (paths.length === 0) {
+        throw new Error(`no path found from ${start} to ${target}`);
+    }
+    return paths;
+}
+
+// move :: cmd -> Handler of path_set
+// move cmd = \s -> (available_paths (s, cmd), cmd)
+export const move: (cmd: Command) => PathSetHandler = (command: Command) => {
+    return new PathSetHandler((start: Command) => {
+        return [availableShortestPaths(start, command), command];
+    });
+};
+export const accept: PathSetHandler = PathSetHandler.fromPathSet([
+    Path.from(["A"]),
+]);
+// inputCommand :: cmd -> Handler of path_set
+// inputCommand cmd = (move cmd); accept
+export const inputCommand = (command: Command) => move(command).product(accept);
+// inputCommands :: list of cmd -> Handler of path_set
+// inputCommands xs = foldl (x -> y -> x; y) munit xs
+// or
+// inputCommands [] = munit
+// inputCommands (x:xs) = (inputCommand x) ; (inputCommands xs)
+export const inputCommands = (commands: Path) => {
+    return commands.reduce((acc, command) => {
+        return acc.product(inputCommand(command));
+    }, PathSetHandler.unit());
+};
+
+// shortest_path_length :: (handler of PathSet) -> handler of Int
+// shortest_path_length h = \s -> (h s) |> foldl ((x,y) -> min(len x,len y)) INF
+export function shortestPathLength(h: PathSetHandler): Handler<number> {
+    return (s: Command) => {
+        const [paths, last] = h.call(s);
+        return [
+            paths.map((x) => x.length).reduce(
+                (x, y) => Math.min(x, y),
+                Infinity,
+            ),
+            last,
+        ];
+    };
+}
+
+// shortest_path_length has property of shortest path length.
+// shortest_path_length (inputCommands x:xs) s = shortest_path_length (inputCommands [x]) s + shortest_path_length (inputCommands xs) x
+/**
+ * the following assertion is always true.
+ * ```ts
+ * for (const s of ["^", "v", "<", ">", "A"] as Command[]) {
+ *      // any path
+ *     const paths = makeRandomSnapshotOfPath();
+ *     const [x, ...xs] = paths;
+ *     const s1 = shortestPathLengthRaw(inputCommands(Path.from([x])))(s)[0];
+ *     const s2 = shortestPathLengthRaw(inputCommands(Path.from(xs)))(x)[0];
+ *     const s3 = shortestPathLengthRaw(inputCommands(paths))(s)[0];
+ *     assertEquals(
+ *         s1 + s2,
+ *         s3,
+ *         `s: ${s}, x: ${x}, xs: ${xs}, s1: ${s1}, s2: ${s2}, s3: ${s3}`,
+ *    );
+ * }
+ */
+
+function double(xs: Path) {
+    return (s1: Command) => {
+        return (s2: Command) => {
+            const [pathset] = inputCommands(xs).call(s1);
+            return pathset.flatMap((p) => {
+                const [pathset2] = inputCommands(p).call(s2);
+                return pathset2;
+            });
+        };
+    };
+}
+// const data = double(Path.from("<A^A>^^AvvvA".split("") as Command[]))("A")("A");
+// console.log(data.map((x) => x.toString()));
+// double xs = \s1 -> \s2 -> flatMap (\cmd -> fst.inputCommands cmd s2) (fst.inputCommands xs s1)
+// = \s1 -> \s2 ->  flatMap ((\cmd -> fst.inputCommands cmd s2) (fst.inputCommands x:xs s1))
+// = \s1 -> \s2 ->  flatMap (\cmd -> fst.inputCommands cmd s2) ((fst.inputCommand x; fst.inputCommands xs) s1)
+// = \s1 -> \s2 ->  flatMap (\cmd -> fst.inputCommands cmd s2) ((move x; accept; fst.inputCommands xs) s1)
+// = \s1 -> \s2 ->  flatMap (\cmd -> fst.inputCommands cmd s2) (
+//      let (p,s3) = (move x; accept; s1) in
+//      p \product (inputCommands xs s3)
+// )
+// = \s1 -> \s2 ->  flatMap (\cmd -> fst.inputCommands cmd s2) (
+//      let (p,s3) = (concat (move x s1) 'A') in
+//      p \product (inputCommands xs s3)
+// )
+// = \s1 -> \s2 -> (
+//      let (p,s3) = (concat (move x s1) 'A') in
+//      flatMap (\cmd -> fst.inputCommands cmd s2) p \product flatMap (\cmd -> fst.inputCommands cmd s2) (fst.inputCommands xs s3)
+// )
+//
+
+// shortest_path_length \s' -> (double xs 'A' s') 'A'
+// = shortest_path_length \s' -> (
+//      let (p,s3) = (concat (move x 'A') 'A') in
+//      flatMap (\cmd -> fst.inputCommands cmd s') p \product flatMap (\cmd -> fst.inputCommands cmd 's) (fst.inputCommands xs s3)
+// ) 'A'
+// = (
+//      let (p,s3) = (concat (move x 'A') 'A') in
+//      let paths = flatMap (\cmd -> fst.inputCommands cmd 'A') p \product flatMap (\cmd -> fst.inputCommands cmd 'A') (fst.inputCommands xs s3)
+//      in
+//      paths |> foldl ((x,y) -> min(len x,len y)) INF
+// )
+// = (
+//      let handler2 = \cmd -> fst.inputCommands cmd 'A' in
+//      let (p,s3) = (concat (move x 'A') 'A') in
+//      let paths = flatMap handler2 p \product flatMap handler2 (fst.inputCommands xs s3) in
+//      paths |> set.min (\x -> len x)
+// )
+// = (
+//     let handler2 = \cmd -> fst.inputCommands cmd 'A' in
+//     let (p,s3) = (concat (move x 'A') 'A') in
+//     set.min (\x -> len x)
+//       (flatMap handler2 p \product flatMap handler2 (fst.inputCommands xs s3))
+// = (
+//     let handler2 = \cmd -> fst.inputCommands cmd 'A' in
+//     let (p,s3) = (concat (move x 'A') 'A') in
+//     (set.min (\x -> len x)
+//         (flatMap handler2 p)) +
+//     (set.min (\x -> len x) (flatMap handler2 (fst.inputCommands xs s3))
+// )
+
+function shortL(
+    curPos: Command,
+    command: Command,
+): number {
+    const [n] = shortestPathLength(inputCommands(Path.from([command])))(curPos);
+    return n;
+}
+
+const memo = new Map<string, number>();
+function myShortestPathLength(
+    depth: number,
+    cmds: Path,
+): number {
+    const key = cmds.toString() + depth;
+    if (memo.has(key)) {
+        return memo.get(key)!;
+    }
+    if (depth === 1) {
+        const ret = shortestPathLength(inputCommands(cmds))("A")[0];
+        memo.set(key, ret);
+        return ret;
+    }
+    const ret = cmds.reduce((acc, cmd) => {
+        const handler = inputCommand(cmd);
+        const [pathset, last] = handler.call(acc.last);
+        const n = pathset.map((x) => myShortestPathLength(depth - 1, x)).reduce(
+            (x, y) => Math.min(x, y),
+            Infinity,
+        );
+        return {
+            last: last,
+            sum: acc.sum + n,
+        };
+    }, {
+        last: "A" as Command,
+        sum: 0,
+    }).sum;
+    memo.set(key, ret);
+    return ret;
+}
+
+if (import.meta.main) {
+    const example = Path.from("<A^A>^^AvvvA".split("") as Command[]);
+    console.log(myShortestPathLength(1, example));
+
+    const codes = Deno.readTextFileSync("input.txt").replaceAll("\r", "").split(
+        "\n",
+    );
+
+    let total = 0;
+    for (const code of codes) {
+        let start = NUMPAD_FIRST_POS;
+        for (const num of code.split("")) {
+            console.log("num", num);
+            const numpadPaths = shortestPathesOnNumpadRobot({ pos: start }, num)
+                .map((p) => p.concat("A"));
+            if (numpadPaths.length === 0) {
+                throw new Error(`No path found for ${num} from ${start}`);
+            }
+            console.log("avail path", numpadPaths.map((p) => p.join("")));
+            
+            let min = Infinity;
+            for (const path of numpadPaths) {
+                const len = myShortestPathLength(25, Path.from(path));
+                min = Math.min(min, len);
+            }
+            const n = parseInt(code.slice(0, 3));
+            console.log("len", min, "n", n);
+            total += n * min;
+
+            start = getPosFromNumpadNumber(num);
+        }
+    }
+    console.log("total", total);
+}