The challenge

In a grid of 4 by 4 squares you want to place a skyscraper in each square with only some clues:

• The height of the skyscrapers is between 1 and 4
• No two skyscrapers in a row or column may have the same number of floors
• A clue is the number of skyscrapers that you can see in a row or column from the outside
• Higher skyscrapers block the view of lower skyscrapers located behind them

Can you write a program that can solve this puzzle?

Example:

To understand how the puzzle works, this is an example of a row with 2 clues. Seen from the left side there are 4 buildings visible while seen from the right side only 1:

4 1

There is only one way in which the skyscrapers can be placed. From left-to-right all four buildings must be visible and no building may hide behind another building:

4 1 2 3 4 1

Example of a 4 by 4 puzzle with the solution:

1 2
2
1
3
1 2
2 1 4 3
3 4 1 2 2
1 4 2 3 1
1 3 2 4
3

• Finish:
 ``````1 `````` ``````public static int[][] solvePuzzle (int[] clues) ``````
• Pass the clues in an array of 16 items. This array contains the clues around the clock, index:
0 1   2   3   15         4 14         5 13         6 12         7  1110 9 8
• If no clue is available, add value `0`
• Each puzzle has only one possible solution
• `SolvePuzzle()` returns matrix `int[][]`. The first indexer is for the row, the second indexer for the column. (Python: returns 4-tuple of 4-tuples, Ruby: 4-Array of 4-Arrays)

The solution in Java code

Option 1:

 `````` 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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 `````` ``````import java.util.Arrays; import java.util.HashSet; import java.util.Set; import java.util.stream.Collectors; import java.util.stream.IntStream; public class SkyScrapers { private final static Integer SIZE = 4; static int[][] solvePuzzle (int[] clues) { return new SkyScrapers().solve(clues); } private static class AbortException extends Exception { } private Table table; private int[][] solve(int[] clues) { table = new Table(clues); try { Position initPosition = table.getInitPosition(); if (iter(initPosition)) return table.table; else throw new RuntimeException("no solution"); } catch (AbortException e) { return table.table; } } private boolean iter(Position position) { Set possibleValues = table.getPossibleValuesForPosition(position); for (Integer possibleValue : possibleValues) { table.setValueForPosition(position, possibleValue); try { Position nextPosition = table.getNextPosition(position); if(iter(nextPosition)) return true; else table.setValueForPosition(position, 0); } catch (AbortException e) { return true; } } return false; } static class Position { Integer x; Integer y; Position(Integer x, Integer y) { this.x = x; this.y = y; } Integer getX() { return x; } Integer getY() { return y; } } static class Table { int[][] table; Line[] horizontalLines = new Line[SIZE]; Line[] verticalLines = new Line[SIZE]; Table(int[] clues) { initTable(); initLinesWithClues(clues); } private Integer getValueForPosition(int x, int y) { return table[y][x]; // inverted! } void setValueForPosition(Position position, int value) { int x = position.getX(); int y = position.getY(); setValueForPosition(x, y, value); verticalLines[x].line[y] = value; horizontalLines[y].line[x] = value; } private void setValueForPosition(int x, int y, int value) { table[y][x] = value; // inverted! } Position getInitPosition() throws AbortException { Position position = new Position(0,0); if (getValueForPosition(0,0) != 0) position = getNextPosition(position); return position; } Position getNextPosition(Position position) throws AbortException { int x = position.getX(); int y = position.getY(); do { if (x == SIZE-1) if (y == SIZE-1) throw new AbortException(); else { x = 0; y++; } else x++; } while (getValueForPosition(x,y) != 0); return new Position(x,y); } Set getPossibleValuesForPosition(Position position) { int x = position.getX(); int y = position.getY(); Set possibleValues = new HashSet<>(); Set possibleValuesForHorizontalLine = horizontalLines[y].getPossibleValuesForPosition(x); Set possibleValuesForVerticalLine = verticalLines[x].getPossibleValuesForPosition(y); possibleValues.addAll(possibleValuesForHorizontalLine); possibleValues.retainAll(possibleValuesForVerticalLine); return possibleValues; } private void initLinesWithClues(int[] clues) { IntStream.range(0, SIZE).forEach(x -> verticalLines[x] = new Line(getVerticalLine(x), clues[x], clues[3*SIZE-1-x])); synchTableWithVerticalLines(); IntStream.range(0, SIZE).forEach(y -> horizontalLines[y] = new Line(getHorizontalLine(y), clues[4*SIZE-1-y], clues[SIZE+y])); synchTableWithHorizontalLines(); } private void synchTableWithHorizontalLines() { for (int x = 0 ; x < SIZE ; x++) for (int y = 0 ; y < SIZE ; y++) setValueForPosition(x, y, horizontalLines[y].line[x]); } private void synchTableWithVerticalLines() { for (int x = 0 ; x < SIZE ; x++) for (int y = 0 ; y < SIZE ; y++) setValueForPosition(x, y, verticalLines[x].line[y]); } private int[] getVerticalLine(int x) { int[] line = new int[SIZE]; IntStream.range(0, SIZE).forEach(y -> line[y] = getValueForPosition(x,y)); return line; } private int[] getHorizontalLine(int y) { int[] line = new int[SIZE]; IntStream.range(0, SIZE).forEach(x -> line[x] = getValueForPosition(x,y)); return line; } private void initTable() { table = new int[SIZE][SIZE]; IntStream.range(0, SIZE).forEach(i -> Arrays.fill(table[i], 0)); } } static class Line { int[] line; int topClue; int bottomClue; private Set allNonInitClues = IntStream.rangeClosed(2, SIZE-1).boxed().collect(Collectors.toSet()); Line(int[] line, int topClue, int bottomClue) { this.line = line; this.topClue = topClue; this.bottomClue = bottomClue; resolveInitClues(); } private void resolveInitClues() { if (topClue == 1) line[0] = SIZE; if (bottomClue == 1) line[SIZE-1] = SIZE; if (topClue == SIZE) IntStream.range(0, SIZE).forEach(i -> line[i] = i + 1); if (bottomClue == SIZE) IntStream.range(0, SIZE).forEach(i -> line[SIZE-1-i] = i + 1); } Set getPossibleValuesForPosition(Integer positionInLine) { if(line[positionInLine] != 0) throw new IllegalArgumentException("Fatal Error : Trying to discover already discovered value !!" + "\n position = " + positionInLine + "\n line : " + this); Set alreadyPlacedValues = getAlreadyPlacedValues(positionInLine); Set possibleValues = IntStream.rangeClosed(1, SIZE).boxed().collect(Collectors.toSet()); possibleValues.removeAll(alreadyPlacedValues); possibleValues.removeIf(valueToTest -> !isCompatibleWithClues(valueToTest, positionInLine)); return possibleValues; } private boolean isCompatibleWithClues(Integer valueToTest, int position) { line[position] = valueToTest; boolean result = true; if(this.isComplete()) result = this.isCompatibleWithClues(); // double negation! line[position] = 0; return result; } private boolean isComplete() { return IntStream.range(0, SIZE).allMatch(i -> line[i] != 0); } private boolean isCompatibleWithClues() { boolean isNotCompatibleWithClues = (allNonInitClues.contains(topClue) && getNumberOfSkyCrapersOnLine(true) != topClue) || (allNonInitClues.contains(bottomClue) && getNumberOfSkyCrapersOnLine(false) != bottomClue); return !isNotCompatibleWithClues; } private int getNumberOfSkyCrapersOnLine(boolean isAscending) { int result = 0; int maxValue = 0; for (int i = 0 ; i < SIZE ; i++) { int positionInLine = isAscending ? i : SIZE-1-i; if (line[positionInLine] > maxValue) { maxValue = line[positionInLine]; result++; } } return result; } private Set getAlreadyPlacedValues(int positionInLine) { Set alreadyPlacedValues = new HashSet(); for (int i = 0 ; i < SIZE ; i ++) { if (i == positionInLine) continue; if (line[i] != 0) alreadyPlacedValues.add(line[i]); } return alreadyPlacedValues; } } } ``````

Option 2:

 `````` 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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 `````` ``````import java.util.*; public class SkyScrapers { public static final int SIZE = 4; public static final int[][] direction = new int[][]{{0,1}, {1,0}, {0,-1}, {-1, 0}}; public static int[][][] square3d; public static int row = 0, col = 0; public static int count; static int[][] solvePuzzle (int[] clues) { generateEmptySquare(); while(count < SIZE * SIZE) { for(int i = 0; i < SIZE * 4; i++) { System.out.printf("i = %d, clue[i]=%d, r = %d, c = %d%n", i, clues[i], row, col); cleanAllHintsInLine(i); checkLineForOneHint(i); useClue(i, clues[i]); if((i + 1) % SIZE != 0) { row += direction[i / SIZE][0]; col += direction[i / SIZE][1]; } } checkSquareForOneHint(); } int[][] result = new int[SIZE][SIZE]; for(int i = 0; i < SIZE; i++) { for(int j = 0; j < SIZE; j++) { result[i][j] = square3d[i][j][0]; } } return result; } static void useClue(int i, int clue) { switch(clue) { case 1: setSkyscraper(iRow(i, 0), iCol(i, 0), SIZE); cleanHintLine(i, SIZE); break; case 2: cleanHint(i, 0, SIZE); if(getSkyscraper(i, SIZE - 1) == SIZE) { setSkyscraper(iRow(i, 0), iCol(i, 0), SIZE - 1); } if(getSkyscraper(i, SIZE - 2) == SIZE) { cleanHint(i, 1, SIZE - 1); } if(getSkyscraper(i, SIZE - 2) == SIZE && getSkyscraper(i, SIZE - 1) == SIZE - 1) { setSkyscraper(iRow(i, 0), iCol(i, 0), 2); setSkyscraper(iRow(i, 1), iCol(i, 1), 1); } break; case 3: cleanHint(i, 0, SIZE); cleanHint(i, 1, SIZE); cleanHint(i, 0, SIZE - 1); if(getSkyscraper(i, SIZE - 1) == SIZE && getSkyscraper(i, SIZE - 2) == SIZE - 1) { setSkyscraper(iRow(i, 0), iCol(i, 0), 2); setSkyscraper(iRow(i, 1), iCol(i, 1), 1); } else if (getSkyscraper(i, SIZE - 1) == SIZE - 1 && getSkyscraper(i, SIZE - 2) == SIZE) { setSkyscraper(iRow(i, 0), iCol(i, 0), 1); setSkyscraper(iRow(i, 1), iCol(i, 1), 2); } else if (getSkyscraper(i, SIZE - 2) == SIZE && getSkyscraper(i, 0) == 2) { setSkyscraper(iRow(i, 1), iCol(i, 1), 3); } break; case SIZE: for(int j = 0; j < SIZE; j++) { setSkyscraper(iRow(i, j), iCol(i, j), j + 1); } break; } } static void generateEmptySquare() { square3d = new int[SIZE][SIZE][SIZE+1]; for(int r = 0; r < SIZE; r++) { for(int c = 0; c < SIZE; c++) { for(int d = 0; d < SIZE + 1; d++) { square3d[r][c][d] = d; } } } count = 0; } static void checkSquareForOneHint() { for(int i = 0; i < SIZE; i++) { for(int j = 0; j < SIZE; j++) { checkIsOnlyOneHintAndSetIfYes(i, j); } } } static void checkLineForOneHint(int i) { for(int h = 1; h <= SIZE; h++) { int pos = 0; int count = 0; for(int j = 0; j < SIZE; j++) { if(square3d[iRow(i, j)][iCol(i, j)][h] > 0) { count++; pos = j; } } if(count == 1) { setSkyscraper(iRow(i, pos), iCol(i, pos), h); } } } static boolean checkIsOnlyOneHintAndSetIfYes(int row, int col) { if(square3d[row][col][0] > 0) { return true; } int num = 0; int count = 0; for(int dig = 1; dig <= SIZE; dig++) { if(square3d[row][col][dig] > 0) { count++; num = dig; } } if(count == 1) { setSkyscraper(row, col, num); return true; } return false; } static void setSkyscraper(int row, int col, int value) { if(square3d[row][col][0] == 0) { square3d[row][col][0] = value; for(int h = 0; h < SIZE; h++) { square3d[row][col][h+1] = 0; square3d[row][h][value] = 0; square3d[h][col][value] = 0; } count++; } } static int getSkyscraper(int i, int j) { return square3d[iRow(i, j)][iCol(i, j)][0]; } static int iRow(int i, int j) { return row + direction[((i / SIZE) + 1) % SIZE][0] * j; } static int iCol(int i, int j) { return col + direction[((i / SIZE) + 1) % SIZE][1] * j; } static int howManySee(int i) { int max = 0; int count = 0; for(int j = 0; j < SIZE; j++) { if(square3d[iRow(i, j)][iCol(i, j)][0] > max) { max = square3d[iRow(i, j)][iCol(i, j)][0]; count++; } } return count; } static void cleanAllHintsInLine(int i) { for(int j=0; j < SIZE; j++) { int v = getSkyscraper(i, j); if(v > 0) { cleanHintLine(i, v); } } } static void cleanHintLine(int i, int hint) { for(int j = 0; j < SIZE; j++) { cleanHint(i, j, hint); } } static void cleanHint(int i, int j, int hint) { square3d[iRow(i, j)][iCol(i, j)][hint] = 0; } } ``````

Test cases to validate our solution

 `````` 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 `````` ``````import org.junit.Test; import static org.junit.Assert.assertEquals; import org.junit.runners.JUnit4; public class SolutionTest { private static int clues[][] = { { 2, 2, 1, 3, 2, 2, 3, 1, 1, 2, 2, 3, 3, 2, 1, 3 }, { 0, 0, 1, 2, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0 } }; private static int outcomes[][][] = { { { 1, 3, 4, 2 }, { 4, 2, 1, 3 }, { 3, 4, 2, 1 }, { 2, 1, 3, 4 } }, { { 2, 1, 4, 3 }, { 3, 4, 1, 2 }, { 4, 2, 3, 1 }, { 1, 3, 2, 4 } } }; @Test public void testSolvePuzzle1 () { assertEquals (SkyScrapers.solvePuzzle (clues[0]), outcomes[0]); } @Test public void testSolvePuzzle2 () { assertEquals (SkyScrapers.solvePuzzle (clues[1]), outcomes[1]); } } ``````