puzzle.js

#!/usr/bin/env node

/**
 * Solve the problem by guessing.
 * 
 * @param {Int8Array} src 
 * @param {Int8Array} dst 
 * @param {() => void} fn 
 */
function solve(src, dst, fn) {
    if (src.length == 0) {
        fn();
        return;
    }

    let tmp = new Int8Array(src.length - 1);
    src.forEach((x) => {
        dst[0] = x;
        let k = 0;
        src.forEach((y, j) => {
            if (x != y) {
                tmp[k] = y;
                k++;
            }
        });
        solve(tmp, dst.subarray(1), fn);
    });
}

/**
 * Create a string to show the solution.
 * 
 * @param {Int8Array} sol 
 */
function format(sol) {
    let ad = sol[1]*10 + sol[2],
        bd = sol[4]*10 + sol[5],
        cd = sol[7]*10 + sol[8];
    return `${sol[0]}/${ad} + ${sol[3]}/${bd} + ${sol[6]}/${cd} = 1`;
}

/**
 * Does this guess solve the problem?
 * 
 * @param {Int8Array} sol 
 * @returns {boolean}
 */
function isSolution(sol) {
    let ad = sol[1]*10 + sol[2],
        bd = sol[4]*10 + sol[5],
        cd = sol[7]*10 + sol[8],
        s = sol[0]/ad + sol[3]/bd + sol[6]/cd;
    return s > 0.9999 && s < 1.0001;
    
}

let n = 9,
    src = new Int8Array(n).map((x, i) => {
        return i + 1;
    }),
    dst = new Int8Array(n);

solve(src, dst, () => {
    if (isSolution(dst)) {
        console.log(format(dst));
    }
});