project euler 61

Problem 61

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Cyclical figurate numbers

Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:

   TriangleP3,n=n(n+1)/21, 3, 6, 10, 15, …SquareP4,n=n21, 4, 9, 16, 25, …PentagonalP5,n=n(3n−1)/21, 5, 12, 22, 35, …HexagonalP6,n=n(2n−1)1, 6, 15, 28, 45, …HeptagonalP7,n=n(5n−3)/21, 7, 18, 34, 55, …OctagonalP8,n=n(3n−2)1, 8, 21, 40, 65, …

The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.

1. The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).
2. Each polygonal type: triangle (P3,127=8128), square (P4,91=8281), and pentagonal (P5,44=2882), is represented by a different number in the set.
3. This is the only set of 4-digit numbers with this property.

Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set.

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循环的多边形数

三角形数、正方形数、五边形数、六边形数、七边形数和八边形数统称为多边形数。它们分别由如下的公式给出：

   三角形数P3,n=n(n+1)/21, 3, 6, 10, 15, …正方形数P4,n=n21, 4, 9, 16, 25, …五边形数P5,n=n(3n−1)/21, 5, 12, 22, 35, …六边形数P6,n=n(2n−1)1, 6, 15, 28, 45, …七边形数P7,n=n(5n−3)/21, 7, 18, 34, 55, …八边形数P8,n=n(3n−2)1, 8, 21, 40, 65, …

由三个4位数8128、2882、8281构成的有序集有如下三个有趣的性质。

1. 这个集合是循环的，每个数的后两位是后一个数的前两位（最后一个数的后两位也是第一个数的前两位）。
2. 每种多边形数——三角形数（P3,127=8128）、正方形数（P4,91=8281）和五边形数（P5,44=2882）——在其中各有一个代表。
3. 这是唯一一个满足上述性质的4位数有序集。

存在唯一一个包含六个4位数的有序循环集，每种多边形数——三角形数、正方形数、五边形数、六边形数、七边形数和八边形数——在其中各有一个代表。求这个集合的元素和。

package projecteuler;import java.util.ArrayList;import java.util.Arrays;import java.util.HashMap;import java.util.List;import java.util.Map;import java.util.Map.Entry;import junit.framework.TestCase;public class Prj61 extends TestCase {public static int NUM_OF_SIZE = 6;public void testCyclicalFigurateNumbers() {DfsCyclicalFigurateNumbers2 dfsC2 = new DfsCyclicalFigurateNumbers2();dfsC2.doTest();/** * 分层搜索,计算量太大,改为起始点固定，一步一步只要保持前一个最后两位和后一个量前两位相同即可.(只需保持一个使用类型标识,下次选取不同即可 * ，减小搜索深度和复杂度) */// DfsCyclicalFigurateNumbers dfsC = new DfsCyclicalFigurateNumbers();// dfsC.doTest();}public interface ICalculateDataList {public List<Integer> getDataList(int start, int end);}public static class CyclicalFigurateNumbersTestCase2 {public int n;public int[] data;public int[] idx;public CyclicalFigurateNumbersTestCase2(int n) {super();this.n = n;data = new int[n];idx = new int[n];}public void setValue(int id, int val, int mapId) {data[id] = val;idx[id] = mapId;}public int[] getResult() {return Arrays.copyOf(data, data.length);}public boolean isOutOfBoundary(int id) {return id >= data.length;}public boolean isFind() {int[] _front = splitInt(data[0]);int[] _tail = splitInt(data[NUM_OF_SIZE - 1]);return _front[0] == _tail[1];}public boolean isValidMap(int id, int mapId) {for (int i = 0; i < id; i++) {if (idx[i] == mapId) {return false;}}return true;}public boolean isValid(int id, int val) {if (id == 0) {return true;}int[] _front = splitInt(data[id - 1]);int[] _tail = splitInt(val);return _front[1] == _tail[0];}public void printResult() {for (int i = 0; i < data.length; i++) {System.out.print(data[i] + ",");}System.out.println();for (int i = 0; i < idx.length; i++) {System.out.print(idx[i] + ",");}System.out.println();}}public static class DfsCyclicalFigurateNumbers2 {public List<int[]> rsList = new ArrayList<int[]>();public void doTest() {ICalculateDataList Octagonal = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * (3 * n - 2);if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};ICalculateDataList Heptagonal = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * (5 * n - 3) / 2;if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};ICalculateDataList Hexagonal = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * (2 * n - 1);if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};ICalculateDataList Pentagonal = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * (3 * n - 1) / 2;if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};ICalculateDataList Square = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * n;if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};ICalculateDataList Triangle = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * (n + 1) / 2;if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};Map<Integer, List<Integer>> mapList = new HashMap<Integer, List<Integer>>();mapList.put(0, Octagonal.getDataList(1000, 9999));mapList.put(1, Heptagonal.getDataList(1000, 9999));mapList.put(2, Hexagonal.getDataList(1000, 9999));mapList.put(3, Pentagonal.getDataList(1000, 9999));mapList.put(4, Square.getDataList(1000, 9999));mapList.put(5, Triangle.getDataList(1000, 9999));// mapList.put(0, Pentagonal.getDataList(1000, 9999));// mapList.put(1, Square.getDataList(1000, 9999));// mapList.put(2, Triangle.getDataList(1000, 9999));CyclicalFigurateNumbersTestCase2 ts = new CyclicalFigurateNumbersTestCase2(NUM_OF_SIZE);dfs(0, ts, mapList, rsList);System.out.println("result=_____________");for (int i = 0; i < rsList.size(); i++) {int[] rs = rsList.get(i);for (int j = 0; j < rs.length; j++) {System.out.print(rs[j] + ",");}System.out.println();}}private void dfs(int startIndex, CyclicalFigurateNumbersTestCase2 ts,Map<Integer, List<Integer>> mapList, List<int[]> rs) {if (ts.isOutOfBoundary(startIndex)) {if (ts.isFind()) {rs.add(ts.getResult());ts.printResult();}return;}if (startIndex == 0) {List<Integer> primeList = mapList.get(startIndex);for (int i = 0; i < primeList.size(); i++) {int setVal = primeList.get(i);if (ts.isValid(startIndex, setVal)) {ts.setValue(startIndex, setVal, 0);dfs(startIndex + 1, ts, mapList, rs);}}} else {for (Entry<Integer, List<Integer>> entry : mapList.entrySet()) {if (!ts.isValidMap(startIndex, entry.getKey())) {continue;}List<Integer> primeList = entry.getValue();for (int i = 0; i < primeList.size(); i++) {int setVal = primeList.get(i);if (ts.isValid(startIndex, setVal)) {ts.setValue(startIndex, setVal, entry.getKey());dfs(startIndex + 1, ts, mapList, rs);}}}}}}public static class CyclicalFigurateNumbersTestCase {public int n;public int[] data;public CyclicalFigurateNumbersTestCase(int n) {super();this.n = n;data = new int[n];}public void setValue(int id, int val) {data[id] = val;}public int[] getResult() {return Arrays.copyOf(data, data.length);}public boolean isOutOfBoundary(int id) {return id >= data.length;}public boolean isFind() {Combination cb = new Combination(data.length, 0);List<int[]> allCbs = cb.generateCom();for (int i = 0; i < allCbs.size(); i++) {int[] arr = allCbs.get(i);int[] tp = new int[arr.length];for (int j = 0; j < arr.length; j++) {tp[j] = data[arr[j]];}if (isCheckOK(tp)) {return true;}}return false;}private boolean isCheckOK(int[] tp) {for (int i = 0; i < tp.length - 1; i++) {int[] _front = splitInt(tp[i]);int[] _tail = splitInt(tp[i + 1]);if (_tail[0] != _front[1]) {return false;}}int[] _front = splitInt(tp[0]);int[] _tail = splitInt(tp[tp.length - 1]);return _front[0] == _tail[1];}public boolean isValid(int id, int val) {return true;}public void printResult() {for (int i = 0; i < data.length; i++) {System.out.print(data[i] + ",");}System.out.println();}}public static class DfsCyclicalFigurateNumbers {public List<int[]> rsList = new ArrayList<int[]>();public void doTest() {ICalculateDataList Octagonal = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * (3 * n - 2);if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};ICalculateDataList Heptagonal = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * (5 * n - 3) / 2;if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};ICalculateDataList Hexagonal = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * (2 * n - 1);if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};ICalculateDataList Pentagonal = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * (3 * n - 1) / 2;if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};ICalculateDataList Square = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * n;if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};ICalculateDataList Triangle = new ICalculateDataList() {@Overridepublic List<Integer> getDataList(int start, int end) {List<Integer> dataList = new ArrayList<Integer>();for (int n = 1;; n++) {int val = n * (n + 1) / 2;if (val >= start) {if (val > end) {break;}int[] _val = splitInt(val);if (_val[0] >= 10 && _val[1] >= 10) {dataList.add(val);}}}return dataList;}};Map<Integer, List<Integer>> mapList = new HashMap<Integer, List<Integer>>();mapList.put(0, Octagonal.getDataList(1000, 9999));mapList.put(1, Heptagonal.getDataList(1000, 9999));mapList.put(2, Hexagonal.getDataList(1000, 9999));mapList.put(3, Pentagonal.getDataList(1000, 9999));mapList.put(4, Square.getDataList(1000, 9999));mapList.put(5, Triangle.getDataList(1000, 9999));// mapList.put(0, Pentagonal.getDataList(1000, 9999));// mapList.put(1, Square.getDataList(1000, 9999));// mapList.put(2, Triangle.getDataList(1000, 9999));CyclicalFigurateNumbersTestCase ts = new CyclicalFigurateNumbersTestCase(NUM_OF_SIZE);dfs(0, ts, mapList, rsList);System.out.println("result=_____________");for (int i = 0; i < rsList.size(); i++) {int[] rs = rsList.get(i);for (int j = 0; j < rs.length; j++) {System.out.print(rs[j] + ",");}System.out.println();}}private void dfs(int startIndex, CyclicalFigurateNumbersTestCase ts,Map<Integer, List<Integer>> mapList, List<int[]> rs) {if (ts.isOutOfBoundary(startIndex)) {ts.printResult();if (ts.isFind()) {rs.add(ts.getResult());ts.printResult();}return;}List<Integer> primeList = mapList.get(startIndex);for (int i = 0; i < primeList.size(); i++) {int setVal = primeList.get(i);if (ts.isValid(startIndex, setVal)) {ts.setValue(startIndex, setVal);dfs(startIndex + 1, ts, mapList, rs);}}}}public static int[] splitInt(int val) {String str = Integer.toString(val);return new int[] { Integer.parseInt(str.substring(0, 2)),Integer.parseInt(str.substring(2, str.length())) };}public static class Combination {private int[] startArr;public Combination(int size) {this(size, 1);}public Combination(int size, int startNum) {startArr = new int[size];for (int i = 0; i < size; i++) {startArr[i] = i + startNum;}}public List<int[]> generateCom() {List<int[]> ret = new ArrayList<int[]>();ret.add(Arrays.copyOf(startArr, startArr.length));while (true) {int lastAsc = findLastAsc(startArr);if (lastAsc == -1) {break;}int lasBigThanAsc = findBigThanAsc(startArr, lastAsc);exchangeEach(lastAsc, lasBigThanAsc, startArr);ret.add(Arrays.copyOf(startArr, startArr.length));}return ret;}private int findBigThanAsc(int[] startArr2, int lastAsc) {int i = 0;for (i = startArr2.length - 1; i > lastAsc; i--) {if (startArr2[i] > startArr2[lastAsc]) {return i;}}assert (i > lastAsc);return i;}private void exchangeEach(int lastAsc, int lasBigThanAsc,int[] startArr2) {int temp = startArr2[lastAsc];startArr2[lastAsc] = startArr2[lasBigThanAsc];startArr2[lasBigThanAsc] = temp;int[] sortArr = getCopyArr(lastAsc + 1, startArr2);for (int i = 0; i < sortArr.length / 2; i++) {temp = sortArr[sortArr.length - 1 - i];sortArr[sortArr.length - 1 - i] = sortArr[i];sortArr[i] = temp;}for (int i = lastAsc + 1; i < startArr2.length; i++) {startArr2[i] = sortArr[i - lastAsc - 1];}}private int[] getCopyArr(int start, int[] startArr2) {int[] ret = new int[startArr2.length - start];for (int i = start; i < startArr2.length; i++) {ret[i - start] = startArr2[i];}return ret;}private int findLastAsc(int[] startArr2) {for (int i = startArr2.length - 1; i > 0; i--) {if (startArr2[i] > startArr2[i - 1]) {return i - 1;}}return -1;}public long getIntVal(int[] arr) {long sum = arr[0];for (int i = 1; i < arr.length; i++) {sum = sum * 10 + arr[i];}return sum;}}}