Bruteforce/DFS/Backtracking Algorithm using Javascript to Solve
- Time:2020-09-07 12:13:31
- Class:Weblog
- Read:26
Use digit 1 to 8 only once and fill the below grid:

digit-grid
The arithmetic should proceed exactly from left to right and top to the bottom.
Bruteforce Algorithm to Solve the Numbers Grid Puzzle
The bruteforce algorithm should be a very trivial solution. There are 8 numbers to put in the 8 cell. Thus, the total solutions to check is 8! We can use 8 ugly nested loops and skip duplicates. The code nesting isn’t looking good, and is not flexible when the puzzle is later extended e.g. to a much bigger dimension 10×10 etc.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | for (let a = 1; a <= 8; ++ a) { let b = 13 - a; if (b >= 9) continue; if (b == a) continue; for (let c = 1; c <= 8; ++ c) { if ((c == a) || (c == b)) continue; for (let d = 1; d <= 8; ++ d) { if ((d == a) || (d == b) || (d == c)) continue; for (let e = 1; e <= 8; ++ e) { if ((e == a) || (e == b) || (e == c) || (e == d)) continue; if ((c - d) * e == 4) { for (let f = 1; f <= 8; ++ f) { if ((f == a) || (f == b) || (f == c) || (f == d) || (f == e)) continue; for (let g = 1; g <= 8; ++ g) { if ((g == a) || (g == b) || (g == c) || (g == d) || (g == e) || (g == f)) continue; for (let h = 1; h <= 8; ++ h) { if ((h == a) || (h == b) || (h == c) || (h == d) || (h == e) || (h == f) || (h == g)) continue; if ((a + c == 4 * f) && ((b - d) * g == 4) && (9 - e - h == 4)) { console.log(a, "+", b, "-", 9, "=4"); console.log(c, "-", d, "*", e, "=4"); console.log(f, "+", g, "-", h, "=4"); } } } } } } } } } |
for (let a = 1; a <= 8; ++ a) { let b = 13 - a; if (b >= 9) continue; if (b == a) continue; for (let c = 1; c <= 8; ++ c) { if ((c == a) || (c == b)) continue; for (let d = 1; d <= 8; ++ d) { if ((d == a) || (d == b) || (d == c)) continue; for (let e = 1; e <= 8; ++ e) { if ((e == a) || (e == b) || (e == c) || (e == d)) continue; if ((c - d) * e == 4) { for (let f = 1; f <= 8; ++ f) { if ((f == a) || (f == b) || (f == c) || (f == d) || (f == e)) continue; for (let g = 1; g <= 8; ++ g) { if ((g == a) || (g == b) || (g == c) || (g == d) || (g == e) || (g == f)) continue; for (let h = 1; h <= 8; ++ h) { if ((h == a) || (h == b) || (h == c) || (h == d) || (h == e) || (h == f) || (h == g)) continue; if ((a + c == 4 * f) && ((b - d) * g == 4) && (9 - e - h == 4)) { console.log(a, "+", b, "-", 9, "=4"); console.log(c, "-", d, "*", e, "=4"); console.log(f, "+", g, "-", h, "=4"); } } } } } } } } }
The only solution found by the above Javascript bruteforce code is:
5 + 8 - 9 =4 + - - 7 - 6 * 4 =4 / * - 3 + 2 - 1 =4 = = = 4 4 4
Another bruteforce algorithm would be to obtain the 8! permutation and then check the validity of each solution.
DFS with BackTracking Algorithm to Solve the Number Puzzle
A better approach would be to use Depth First Search Algorithm to backtracking the search tree – this allows us to abandon the search branches which can’t be solutions:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 | (function dfs(sol) { if (sol.length == 9) { console.log(sol); return true; } if (sol.length == 2) { if (sol[0] + sol[1] - 9 != 4) return false; sol.push(9); // 9 is given //return dfs(sol); } for (let i = 1; i <= 8; ++ i) { if (!sol.includes(i)) { // can only use once if (sol.length == 5) { // second row if ((sol[3] - sol[4]) * i != 4) continue; } if (sol.length == 6) { // first column if ((sol[0] + sol[3]) / i != 4) continue; } if (sol.length == 7) { // second column if ((sol[1] - sol[4]) * i != 4) continue; } if (sol.length == 8) { // third row if (sol[6] + sol[7] - i != 4) continue; // third column if (sol[2] - sol[5] - i != 4) continue; } sol.push(i); if (dfs(sol)) { return true; } sol.pop(); } } return false; })([]); |
(function dfs(sol) { if (sol.length == 9) { console.log(sol); return true; } if (sol.length == 2) { if (sol[0] + sol[1] - 9 != 4) return false; sol.push(9); // 9 is given //return dfs(sol); } for (let i = 1; i <= 8; ++ i) { if (!sol.includes(i)) { // can only use once if (sol.length == 5) { // second row if ((sol[3] - sol[4]) * i != 4) continue; } if (sol.length == 6) { // first column if ((sol[0] + sol[3]) / i != 4) continue; } if (sol.length == 7) { // second column if ((sol[1] - sol[4]) * i != 4) continue; } if (sol.length == 8) { // third row if (sol[6] + sol[7] - i != 4) continue; // third column if (sol[2] - sol[5] - i != 4) continue; } sol.push(i); if (dfs(sol)) { return true; } sol.pop(); } } return false; })([]);
Running the above Javascript DFS code prints the 8 numbers that is the correct solution:
1 2 3 4 | [ 5, 8, 9, 7, 6, 4, 3, 2, 1 ] |
[ 5, 8, 9, 7, 6, 4, 3, 2, 1 ]
The solution is superior to the above bruteforce approach. This can also be solved via Breadth First Search Algorithm.
–EOF (The Ultimate Computing & Technology Blog) —
Recommend:Five Tips to Lower Your Tax Bill in 2020
Bruteforce Algorithm to Compute the Maxmium Powerful Digit Sum u
4 Frequently Discussed SEO Myths Exposed
3 Reasons Why Graphic Designers Need to Self-Promote through Ins
How to Split a String in C++?
The Best Instagram Feed WordPress Plugins to Use
Trie Class Data Structure in Python
How to Monitor the CPU Temperature of Raspberry PI using Python
Listen to what an SEO expert has to say about its benefits
Depth First Search Algorithm to Compute the Smallest String Star
- Comment list
-
- Comment add