Problem: you have a standard 8×8 chessboard, empty but for a single knight on some square. Your task is to emit a series of legal knight moves that result in the knight visiting every square on the chessboard exactly once. Note that it is not a requirement that the tour be “closed”; that is, the knight need not end within a single move of its start position.
Input and output may be textual or graphical, according to the conventions of the programming environment. If textual, squares should be indicated in algebraic notation. The output should indicate the order in which the knight visits the squares, starting with the initial position. The form of the output may be a diagram of the board with the squares numbered according to visitation sequence, or a textual list of algebraic coordinates in order, or even an actual animation of the knight moving around the chessboard.
Input: starting square
Output: move sequence
Uses Warnsdorff’s rule and (iterative) backtracking if that fails.
#include <iostream> #include <iomanip> #include <array> #include <string> #include <tuple> #include <algorithm> using namespace std; template<int N = 8> class Board { public: array<pair<int, int>, 8> moves; array<array<int, N>, N> data; Board() { moves[0] = make_pair(2, 1); moves[1] = make_pair(1, 2); moves[2] = make_pair(-1, 2); moves[3] = make_pair(-2, 1); moves[4] = make_pair(-2, -1); moves[5] = make_pair(-1, -2); moves[6] = make_pair(1, -2); moves[7] = make_pair(2, -1); } array<int, 8> sortMoves(int x, int y) const { array<tuple<int, int>, 8> counts; for(int i = 0; i < 8; ++i) { int dx = get<0>(moves[i]); int dy = get<1>(moves[i]); int c = 0; for(int j = 0; j < 8; ++j) { int x2 = x + dx + get<0>(moves[j]); int y2 = y + dy + get<1>(moves[j]); if (x2 < 0 || x2 >= N || y2 < 0 || y2 >= N) continue; if(data[y2][x2] != 0) continue; c++; } counts[i] = make_tuple(c, i); } // Shuffle to randomly break ties random_shuffle(counts.begin(), counts.end()); // Lexicographic sort sort(counts.begin(), counts.end()); array<int, 8> out; for(int i = 0; i < 8; ++i) out[i] = get<1>(counts[i]); return out; } void solve(string start) { for(int v = 0; v < N; ++v) for(int u = 0; u < N; ++u) data[v][u] = 0; int x0 = start[0] - 'a'; int y0 = N - (start[1] - '0'); data[y0][x0] = 1; array<tuple<int, int, int, array<int, 8>>, N*N> order; order[0] = make_tuple(x0, y0, 0, sortMoves(x0, y0)); int n = 0; while(n < N*N-1) { int x = get<0>(order[n]); int y = get<1>(order[n]); bool ok = false; for(int i = get<2>(order[n]); i < 8; ++i) { int dx = moves[get<3>(order[n])[i]].first; int dy = moves[get<3>(order[n])[i]].second; if(x+dx < 0 || x+dx >= N || y+dy < 0 || y+dy >= N) continue; if(data[y + dy][x + dx] != 0) continue; ++n; get<2>(order[n]) = i + 1; data[y+dy][x+dx] = n + 1; order[n] = make_tuple(x+dx, y+dy, 0, sortMoves(x+dx, y+dy)); ok = true; break; } if(!ok) // Failed. Backtrack. { data[y][x] = 0; --n; } } } template<int N> friend ostream& operator<<(ostream &out, const Board<N> &b); }; template<int N> ostream& operator<<(ostream &out, const Board<N> &b) { for (int v = 0; v < N; ++v) { for (int u = 0; u < N; ++u) { if (u != 0) out << ","; out << setw(3) << b.data[v][u]; } out << endl; } return out; } int main() { Board<5> b1; b1.solve("c3"); cout << b1 << endl; Board<8> b2; b2.solve("b5"); cout << b2 << endl; Board<31> b3; // Max size for <1000 squares b3.solve("a1"); cout << b3 << endl; return 0; }
Output:
23, 16, 11, 6, 21 10, 5, 22, 17, 12 15, 24, 1, 20, 7 4, 9, 18, 13, 2 25, 14, 3, 8, 19 63, 20, 3, 24, 59, 36, 5, 26 2, 23, 64, 37, 4, 25, 58, 35 19, 62, 21, 50, 55, 60, 27, 6 22, 1, 54, 61, 38, 45, 34, 57 53, 18, 49, 44, 51, 56, 7, 28 12, 15, 52, 39, 46, 31, 42, 33 17, 48, 13, 10, 43, 40, 29, 8 14, 11, 16, 47, 30, 9, 32, 41 275,112, 19,116,277,604, 21,118,823,770, 23,120,961,940, 25,122,943,926, 27,124,917,898, 29,126,911,872, 31,128,197,870, 33 18,115,276,601, 20,117,772,767, 22,119,958,851, 24,121,954,941, 26,123,936,925, 28,125,912,899, 30,127,910,871, 32,129,198 111,274,113,278,605,760,603,822,771,824,769,948,957,960,939,944,953,942,927,916,929,918,897,908,913,900,873,196,875, 34,869 114, 17,600,273,602,775,766,773,768,949,850,959,852,947,952,955,932,937,930,935,924,915,920,905,894,909,882,901,868,199,130 271,110,279,606,759,610,761,776,821,764,825,816,951,956,853,938,945,934,923,928,919,896,893,914,907,904,867,874,195,876, 35 16,581,272,599,280,607,774,765,762,779,950,849,826,815,946,933,854,931,844,857,890,921,906,895,886,883,902,881,200,131,194 109,270,281,580,609,758,611,744,777,820,763,780,817,848,827,808,811,846,855,922,843,858,889,892,903,866,885,192,877, 36,201 282, 15,582,269,598,579,608,757,688,745,778,819,754,783,814,847,828,807,810,845,856,891,842,859,884,887,880,863,202,193,132 267,108,283,578,583,612,689,614,743,756,691,746,781,818,753,784,809,812,829,806,801,840,835,888,865,862,203,878,191,530, 37 14,569,268,585,284,597,576,619,690,687,742,755,692,747,782,813,752,785,802,793,830,805,860,841,836,879,864,529,204,133,190 107,266,285,570,577,584,613,686,615,620,695,684,741,732,711,748,739,794,751,786,803,800,839,834,861,528,837,188,531, 38,205 286, 13,568,265,586,575,596,591,618,685,616,655,696,693,740,733,712,749,738,795,792,831,804,799,838,833,722,527,206,189,134 263,106,287,508,571,590,587,574,621,592,639,694,683,656,731,710,715,734,787,750,737,796,791,832,721,798,207,532,187,474, 39 12,417,264,567,288,509,572,595,588,617,654,657,640,697,680,713,730,709,716,735,788,727,720,797,790,723,526,473,208,135,186 105,262,289,416,507,566,589,512,573,622,593,638,653,682,659,698,679,714,729,708,717,736,789,726,719,472,533,184,475, 40,209 290, 11,418,261,502,415,510,565,594,513,562,641,658,637,652,681,660,699,678,669,728,707,718,675,724,525,704,471,210,185,136 259,104,291,414,419,506,503,514,511,564,623,548,561,642,551,636,651,670,661,700,677,674,725,706,703,534,211,476,183,396, 41 10,331,260,493,292,501,420,495,504,515,498,563,624,549,560,643,662,635,650,671,668,701,676,673,524,705,470,395,212,137,182 103,258,293,330,413,494,505,500,455,496,547,516,485,552,625,550,559,644,663,634,649,672,667,702,535,394,477,180,397, 42,213 294, 9,332,257,492,329,456,421,490,499,458,497,546,517,484,553,626,543,558,645,664,633,648,523,666,469,536,393,220,181,138 255,102,295,328,333,412,491,438,457,454,489,440,459,486,545,518,483,554,627,542,557,646,665,632,537,478,221,398,179,214, 43 8,319,256,335,296,345,326,409,422,439,436,453,488,441,460,451,544,519,482,555,628,541,522,647,468,631,392,219,222,139,178 101,254,297,320,327,334,411,346,437,408,423,368,435,452,487,442,461,450,445,520,481,556,629,538,479,466,399,176,215, 44,165 298, 7,318,253,336,325,344,349,410,347,360,407,424,383,434,427,446,443,462,449,540,521,480,467,630,391,218,223,164,177,140 251,100,303,300,321,316,337,324,343,350,369,382,367,406,425,384,433,428,447,444,463,430,539,390,465,400,175,216,169,166, 45 6,299,252,317,304,301,322,315,348,361,342,359,370,381,366,405,426,385,432,429,448,389,464,401,174,217,224,163,150,141,168 99,250,241,302,235,248,307,338,323,314,351,362,341,358,371,380,365,404,377,386,431,402,173,388,225,160,153,170,167, 46,143 240, 5, 98,249,242,305,234,247,308,339,232,313,352,363,230,357,372,379,228,403,376,387,226,159,154,171,162,149,142,151, 82 63, 2,239, 66, 97,236,243,306,233,246,309,340,231,312,353,364,229,356,373,378,227,158,375,172,161,148,155,152, 83,144, 47 4, 67, 64, 61,238, 69, 96, 59,244, 71, 94, 57,310, 73, 92, 55,354, 75, 90, 53,374, 77, 88, 51,156, 79, 86, 49,146, 81, 84 1, 62, 3, 68, 65, 60,237, 70, 95, 58,245, 72, 93, 56,311, 74, 91, 54,355, 76, 89, 52,157, 78, 87, 50,147, 80, 85, 48,145
Content is available under GNU Free Documentation License 1.2.
