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package class33;
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public class Problem_0279_PerfectSquares {
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// 暴力解
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public static int numSquares1(int n) {
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int res = n, num = 2;
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while (num * num <= n) {
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int a = n / (num * num), b = n % (num * num);
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res = Math.min(res, a + numSquares1(b));
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num++;
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}
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return res;
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}
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// 1 : 1, 4, 9, 16, 25, 36, ...
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// 4 : 7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71, ...
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// 规律解
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// 规律一：个数不超过4
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// 规律二：出现1个的时候，显而易见
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// 规律三：任何数 % 8 == 7，一定是4个
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// 规律四：任何数消去4的因子之后，剩下rest，rest % 8 == 7，一定是4个
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public static int numSquares2(int n) {
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int rest = n;
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while (rest % 4 == 0) {
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rest /= 4;
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}
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if (rest % 8 == 7) {
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return 4;
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}
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int f = (int) Math.sqrt(n);
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if (f * f == n) {
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return 1;
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}
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for (int first = 1; first * first <= n; first++) {
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int second = (int) Math.sqrt(n - first * first);
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if (first * first + second * second == n) {
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return 2;
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}
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}
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return 3;
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}
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// 数学解
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// 1）四平方和定理
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// 2）任何数消掉4的因子，结论不变
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public static int numSquares3(int n) {
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while (n % 4 == 0) {
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n /= 4;
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}
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if (n % 8 == 7) {
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return 4;
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}
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for (int a = 0; a * a <= n; ++a) {
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// a * a + b * b = n
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int b = (int) Math.sqrt(n - a * a);
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if (a * a + b * b == n) {
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return (a > 0 && b > 0) ? 2 : 1;
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}
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}
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return 3;
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}
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public static void main(String[] args) {
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for (int i = 1; i < 1000; i++) {
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System.out.println(i + " , " + numSquares1(i));
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}
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}
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}
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