Ruudukko

//const { performance } = require("perf_hooks")
//const start = performance.now()

// Get input
const [length, ...lines] = require("fs").readFileSync(0, "utf8").trim().split("\n").map(l => l.trim().split(" ").map(Number))

function getRelations(x, y) {
    const val = lines[y][x]
    const relations = []
    // Above
    for (let i = y; i > -1; i--){
        if (lines[i][x] < val) relations.push([x, i])
    }

    // Below
    for (let i = y; i < length; i++){
        if (lines[i][x] < val) relations.push([x, i])
    }

    // Left
    for (let i = x; i > -1; i--){
        if (lines[y][i] < val) relations.push([i, y])
    }

    // Right
    for (let i = x; i < length; i++){
        if (lines[y][i] < val) relations.push([i, y])
    }

    return relations
}

function getPaths(relations, all, visited = []) {
    let pathCount = relations.length // Every path of 0 length is still a path (right?)
    if (visited.length === 0) ++pathCount

    for (const [x, y] of relations) {
        if (visited.includes(`${x}-${y}`)) continue
        const nVisited = visited.slice(0)
        nVisited.push(`${x}-${y}`)
        pathCount += getPaths(all[`${x}-${y}`], all, nVisited)
    }
    return pathCount
}

// Build relational object
const relations = Object.fromEntries(lines
    .map((line, y) => line.map((val, x) => ({
        val,
        x,
        y,
        relations: getRelations(x, y)
    })))
    .flat(1)
    .map((obj) => [`${obj.x}-${obj.y}`, obj.relations]))

// Calculate paths
const paths = Object.keys(relations)
    .map((point, ) => getPaths(relations[point], relations))
    .reduce((a, b) => a + b)

console.log(paths)
//console.log("Took", (performance.now() - start) / 1000 + "s")

Test 1

Group: 1, 2, 3

Verdict: ACCEPTED

Test 2

Group: 1, 2, 3

Verdict: ACCEPTED

Test 3

Group: 1, 2, 3

Verdict: ACCEPTED

Test 4

Group: 2, 3

Verdict: ACCEPTED

Test 5

Group: 2, 3

Verdict: TIME LIMIT EXCEEDED

Test 6

Group: 2, 3

Verdict: TIME LIMIT EXCEEDED

Test 7

Group: 3

Verdict: TIME LIMIT EXCEEDED

Test 8

Group: 3

Verdict: TIME LIMIT EXCEEDED

Test 9

Group: 3

Verdict: TIME LIMIT EXCEEDED