题目

多少正方形？

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枚举

设每个横纵相邻点得间距为1，则我们可以依次输出下列边长为$a$的正方形的数量：

边长为1

$a=1$

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边长为$\sqrt{2}$

$a=\sqrt{1^2+1^2}=\sqrt{2}$

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边长为$\sqrt{5}$

$a=\sqrt{1^2+2^2}=\sqrt{5}$

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边长为$\sqrt{8}$

$a=\sqrt{2^2+2^2}=\sqrt{8}$

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边长为$\sqrt{13}$

$a=\sqrt{2^2+3^2}=\sqrt{13}$

按上述尺寸数得各个大小正方形数量如下：

$$\{\begin{array}{ll}{C}_1=9& a=1\\ C_2=4& a=\sqrt{1^2+1^2}=\sqrt{2}\\ C_3=2& a=\sqrt{1^2+2^2}=\sqrt{5}\\ C_4=4& a=\sqrt{2^2+2^2}=2\sqrt{2}\\ C_5=2& a=\sqrt{2^2+3^2}=\sqrt{13}\end{array}$$

$$\begin{array}{c}\begin{array}{rl}C& =C_1+C_2+C_3+C_4+C_5\\ & =9+4+2+4+2\\ & =21\end{array}\end{array}$$

扩展-使用代码来数

1、定义点对象

        /// <summary>/// 点对象/// </summary>public class CPoint{/// <summary>/// 点横坐标/// </summary>public double X { get; set; }/// <summary>/// 点纵坐标/// </summary>public double Y { get; set; }/// <summary>/// 点相对位置角度/// </summary>public double Angle { get; set; }/// <summary>/// 点编号/// </summary>public int Index { get; set; }public CPoint(int index,double x,double y,double a = 0){this.X = x;this.Y = y;this.Index = index;this.Angle = a;}/// <summary>/// 判断两个点是否是同一个点/// </summary>/// <param name="obj"></param>/// <returns></returns>public override bool Equals(object obj){if (obj == null) return false;if (obj.GetType() != typeof(CPoint)) return false;var val =(CPoint) obj;if(this.X == val.X && this.Y == val.Y){return true;}return false;}/// <summary>/// 格式化点对象/// </summary>/// <returns></returns>public override string ToString(){return $"{Index}.({X},{Y}){Angle}";}/// <summary>/// 50倍绘制圆点/// </summary>/// <returns></returns>public RectangleF DrawCircle(){float size = 10;return new RectangleF((float)(X *50 - size / 2), (float)(Y *50 - size / 2), size, size);}/// <summary>/// 50倍绘制点/// </summary>/// <returns></returns>public PointF DrawPnt(){return new PointF((float)(X * 50), (float)(Y * 50));}}

2、定义正方形对象

        /// <summary>/// 正方形对象/// </summary>public class CSqure{/// <summary>/// 正方形的点/// </summary>public List<CPoint> Points { get; set; }/// <summary>/// 正方形边长/// </summary>public double C { get; set; }/// <summary>/// 正方形中心点横坐标/// </summary>public double CenterX { get; set; }/// <summary>/// 正方形中心点纵坐标/// </summary>public double CenterY { get; set; }public CSqure(CPoint pnt1, CPoint pnt2, CPoint pnt3, CPoint pnt4){if (Points == null){Points = new List<CPoint>();}Points.Add(pnt1);Points.Add(pnt2);Points.Add(pnt3);Points.Add(pnt4);}public PointF[] DrawSqure(){return new PointF[4] { Points[0].DrawPnt(), Points[1].DrawPnt(), Points[2].DrawPnt(), Points[3].DrawPnt() };}public PointF DrawCenter(SizeF size){var center = CalculateCenter();var pnt = center.DrawPnt();return new PointF(pnt.X - size.Width / 2, pnt.Y - size.Height / 2);}/// <summary>/// 判断是否正方形/// </summary>/// <returns></returns>public bool IsSqure(){List<double> distances = new List<double>();// Calculate all pairwise distancesfor (int i = 0; i < Points.Count; i++){for (int j = i + 1; j < Points.Count; j++){distances.Add(Distance(Points[i], Points[j]));}}// 将所有点得距离进行排序distances.Sort();// 第一个距离必须大于0，前四个距离依次相等是边，后两个距离是对象线。四边相等且对角线相等为正方形if (distances[0] > 0 && Equal(distances[0], distances[1]) && Equal(distances[1], distances[2]) && Equal(distances[2], distances[3]) &&Equal(distances[4], distances[5])){C = distances[0]; //计算每个点得顺序List<CPoint> polygons = new List<CPoint>();var center = CalculateCenter();CenterX = center.X;CenterY = center.Y;for (int i = 0; i < Points.Count; i++){var angle = CalculateAngle(center, Points[i]);if (angle < 0){angle += 360;}else if (angle >= 360){angle -= 360;}polygons.Add(new CPoint(Points[i].Index, Points[i].X, Points[i].Y, angle));}Points = polygons.OrderBy(o => o.Angle).ToList();return true;}return false;}/// <summary>/// 计算正方形中心点/// </summary>/// <returns></returns>public CPoint CalculateCenter(){double centerX = Points.Average(o => o.X);double centerY = Points.Average(o => o.Y);return new CPoint(0, centerX, centerY);}/// <summary>/// 计算旋转角/// </summary>/// <param name="center"></param>/// <param name="point"></param>/// <returns></returns>public double CalculateAngle(CPoint center, CPoint point){double deltaY = point.Y - center.Y;double deltaX = point.X - center.X;double angleInRadians = Math.Atan2(deltaY, deltaX);double angleInDegrees = angleInRadians * (180.0 / Math.PI);return angleInDegrees;}/// <summary>/// 判断两个距离是否相等/// </summary>/// <param name="d1"></param>/// <param name="d2"></param>/// <returns></returns>public bool Equal(double d1, double d2){if (Math.Abs(d2 - d1) < 0.000001){return true;}return false;}/// <summary>/// 计算两点之间的距离/// </summary>/// <param name="pnt1"></param>/// <param name="pnt2"></param>/// <returns></returns>public double Distance(CPoint pnt1, CPoint pnt2){double a = pnt1.X - pnt2.X;double b = pnt1.Y - pnt2.Y;double c = Math.Sqrt(a * a + b * b);return c;}/// <summary>/// 判断两个正方形是否是同一个正方形/// </summary>/// <param name="obj"></param>/// <returns></returns>public override bool Equals(object obj){if (obj == null) return false;if (obj.GetType() != typeof(CSqure)) return false;var val = (CSqure)obj;if (this.ToString() == obj.ToString()){return true;}return false;}/// <summary>/// 格式化正方形对象/// </summary>/// <returns></returns>public override string ToString(){var temps = Points.OrderBy(o => o.Index);return string.Join("-", temps.Select(o => o.Index));}/// <summary>/// 根据点获取所有正方形/// </summary>/// <param name="Points"></param>/// <returns></returns>public static List<CSqure> GetAllSqures(List<CPoint> Points){List<CSqure> Squres = new List<CSqure>();for (int i = 0; i < Points.Count(); i++){for (int j = 0; j < Points.Count(); j++){for (int k = 0; k < Points.Count(); k++){for (int l = 0; l < Points.Count(); l++){CSqure squre = new CSqure(Points[i], Points[j], Points[k], Points[l]);if (squre.IsSqure())//判断是不是正方形{bool isExist = false;foreach (var item in Squres){if (item.ToString() == squre.ToString()) //判断是不是已经存在{isExist = true;break;}}if (!isExist)//没有找到过则添加到列表{Squres.Add(squre);}}}}}}Squres = Squres.OrderBy(o => o.C).ThenBy(o => o.CenterY).ThenBy(o => o.CenterX).ToList();return Squres;}}

3、初始化所有点

			//Points.Add(new CPoint(index++, 1, 1, 0));//Points.Add(new CPoint(index++, 1, 2, 0));Points.Add(new CPoint(index++, 1, 3, 0));Points.Add(new CPoint(index++, 1, 4, 0));//Points.Add(new CPoint(index++, 1, 5, 0));//Points.Add(new CPoint(index++, 1, 6, 0));//Points.Add(new CPoint(index++, 2, 1, 0));//Points.Add(new CPoint(index++, 2, 2, 0));Points.Add(new CPoint(index++, 2, 3, 0));Points.Add(new CPoint(index++, 2, 4, 0));//Points.Add(new CPoint(index++, 2, 5, 0));//Points.Add(new CPoint(index++, 2, 6, 0));Points.Add(new CPoint(index++, 3, 1, 0));Points.Add(new CPoint(index++, 3, 2, 0));Points.Add(new CPoint(index++, 3, 3, 0));Points.Add(new CPoint(index++, 3, 4, 0));Points.Add(new CPoint(index++, 3, 5, 0));Points.Add(new CPoint(index++, 3, 6, 0));Points.Add(new CPoint(index++, 4, 1, 0));Points.Add(new CPoint(index++, 4, 2, 0));Points.Add(new CPoint(index++, 4, 3, 0));Points.Add(new CPoint(index++, 4, 4, 0));Points.Add(new CPoint(index++, 4, 5, 0));Points.Add(new CPoint(index++, 4, 6, 0));//Points.Add(new CPoint(index++, 5, 1, 0));//Points.Add(new CPoint(index++, 5, 2, 0));Points.Add(new CPoint(index++, 5, 3, 0));Points.Add(new CPoint(index++, 5, 4, 0));//Points.Add(new CPoint(index++, 5, 5, 0));//Points.Add(new CPoint(index++, 5, 6, 0));//Points.Add(new CPoint(index++, 6, 1, 0));//Points.Add(new CPoint(index++, 6, 2, 0));Points.Add(new CPoint(index++, 6, 3, 0));Points.Add(new CPoint(index++, 6, 4, 0));//Points.Add(new CPoint(index++, 6, 5, 0));//Points.Add(new CPoint(index++, 6, 6, 0));

4、调用

public List<CSqure> Squres = CSqure.GetAllSqures(Points);

完整代码

using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Drawing;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Windows.Forms;namespace WindowsFormsApp1
{public partial class Form1 : Form{/// <summary>/// 点对象/// </summary>public class CPoint{/// <summary>/// 点横坐标/// </summary>public double X { get; set; }/// <summary>/// 点纵坐标/// </summary>public double Y { get; set; }/// <summary>/// 点相对位置角度/// </summary>public double Angle { get; set; }/// <summary>/// 点编号/// </summary>public int Index { get; set; }public CPoint(int index,double x,double y,double a = 0){this.X = x;this.Y = y;this.Index = index;this.Angle = a;}/// <summary>/// 判断两个点是否是同一个点/// </summary>/// <param name="obj"></param>/// <returns></returns>public override bool Equals(object obj){if (obj == null) return false;if (obj.GetType() != typeof(CPoint)) return false;var val =(CPoint) obj;if(this.X == val.X && this.Y == val.Y){return true;}return false;}/// <summary>/// 格式化点对象/// </summary>/// <returns></returns>public override string ToString(){return $"{Index}.({X},{Y}){Angle}";}/// <summary>/// 50倍绘制圆点/// </summary>/// <returns></returns>public RectangleF DrawCircle(){float size = 10;return new RectangleF((float)(X *50 - size / 2), (float)(Y *50 - size / 2), size, size);}/// <summary>/// 50倍绘制点/// </summary>/// <returns></returns>public PointF DrawPnt(){return new PointF((float)(X * 50), (float)(Y * 50));}}/// <summary>/// 正方形对象/// </summary>public class CSqure{/// <summary>/// 正方形的点/// </summary>public List<CPoint> Points { get; set; }/// <summary>/// 正方形边长/// </summary>public double C { get; set; }/// <summary>/// 正方形中心点横坐标/// </summary>public double CenterX { get; set; }/// <summary>/// 正方形中心点纵坐标/// </summary>public double CenterY { get; set; }public CSqure(CPoint pnt1, CPoint pnt2, CPoint pnt3, CPoint pnt4){if (Points == null){Points = new List<CPoint>();}Points.Add(pnt1);Points.Add(pnt2);Points.Add(pnt3);Points.Add(pnt4);}public PointF[] DrawSqure(){return new PointF[4] { Points[0].DrawPnt(), Points[1].DrawPnt(), Points[2].DrawPnt(), Points[3].DrawPnt() };}public PointF DrawCenter(SizeF size){var center = CalculateCenter();var pnt = center.DrawPnt();return new PointF(pnt.X - size.Width / 2, pnt.Y - size.Height / 2);}/// <summary>/// 判断是否正方形/// </summary>/// <returns></returns>public bool IsSqure(){List<double> distances = new List<double>();// Calculate all pairwise distancesfor (int i = 0; i < Points.Count; i++){for (int j = i + 1; j < Points.Count; j++){distances.Add(Distance(Points[i], Points[j]));}}// 将所有点得距离进行排序distances.Sort();// 第一个距离必须大于0，前四个距离依次相等是边，后两个距离是对象线。四边相等且对角线相等为正方形if (distances[0] > 0 && Equal(distances[0], distances[1]) && Equal(distances[1], distances[2]) && Equal(distances[2], distances[3]) &&Equal(distances[4], distances[5])){C = distances[0]; //计算每个点得顺序List<CPoint> polygons = new List<CPoint>();var center = CalculateCenter();CenterX = center.X;CenterY = center.Y;for (int i = 0; i < Points.Count; i++){var angle = CalculateAngle(center, Points[i]);if (angle < 0){angle += 360;}else if (angle >= 360){angle -= 360;}polygons.Add(new CPoint(Points[i].Index, Points[i].X, Points[i].Y, angle));}Points = polygons.OrderBy(o => o.Angle).ToList();return true;}return false;}/// <summary>/// 计算正方形中心点/// </summary>/// <returns></returns>public CPoint CalculateCenter(){double centerX = Points.Average(o => o.X);double centerY = Points.Average(o => o.Y);return new CPoint(0, centerX, centerY);}/// <summary>/// 计算旋转角/// </summary>/// <param name="center"></param>/// <param name="point"></param>/// <returns></returns>public double CalculateAngle(CPoint center, CPoint point){double deltaY = point.Y - center.Y;double deltaX = point.X - center.X;double angleInRadians = Math.Atan2(deltaY, deltaX);double angleInDegrees = angleInRadians * (180.0 / Math.PI);return angleInDegrees;}/// <summary>/// 判断两个距离是否相等/// </summary>/// <param name="d1"></param>/// <param name="d2"></param>/// <returns></returns>public bool Equal(double d1, double d2){if (Math.Abs(d2 - d1) < 0.000001){return true;}return false;}/// <summary>/// 计算两点之间的距离/// </summary>/// <param name="pnt1"></param>/// <param name="pnt2"></param>/// <returns></returns>public double Distance(CPoint pnt1, CPoint pnt2){double a = pnt1.X - pnt2.X;double b = pnt1.Y - pnt2.Y;double c = Math.Sqrt(a * a + b * b);return c;}/// <summary>/// 判断两个正方形是否是同一个正方形/// </summary>/// <param name="obj"></param>/// <returns></returns>public override bool Equals(object obj){if (obj == null) return false;if (obj.GetType() != typeof(CSqure)) return false;var val = (CSqure)obj;if (this.ToString() == obj.ToString()){return true;}return false;}/// <summary>/// 格式化正方形对象/// </summary>/// <returns></returns>public override string ToString(){var temps = Points.OrderBy(o => o.Index);return string.Join("-", temps.Select(o => o.Index));}/// <summary>/// 根据点获取所有正方形/// </summary>/// <param name="Points"></param>/// <returns></returns>public static List<CSqure> GetAllSqures(List<CPoint> Points){List<CSqure> Squres = new List<CSqure>();for (int i = 0; i < Points.Count(); i++){for (int j = 0; j < Points.Count(); j++){for (int k = 0; k < Points.Count(); k++){for (int l = 0; l < Points.Count(); l++){CSqure squre = new CSqure(Points[i], Points[j], Points[k], Points[l]);if (squre.IsSqure())//判断是不是正方形{bool isExist = false;foreach (var item in Squres){if (item.ToString() == squre.ToString()) //判断是不是已经存在{isExist = true;break;}}if (!isExist)//没有找到过则添加到列表{Squres.Add(squre);}}}}}}Squres = Squres.OrderBy(o => o.C).ThenBy(o => o.CenterY).ThenBy(o => o.CenterX).ToList();return Squres;}}public List<CPoint> Points = new List<CPoint>();public List<CSqure> Squres = new List<CSqure>();Timer timer = new Timer();public Form1(){InitializeComponent();this.DoubleBuffered = true;timer.Tick += Timer_Tick;timer.Interval = 600;timer.Start();int index = 1;//Points.Add(new CPoint(index++, 1, 1, 0));//Points.Add(new CPoint(index++, 1, 2, 0));Points.Add(new CPoint(index++, 1, 3, 0));Points.Add(new CPoint(index++, 1, 4, 0));//Points.Add(new CPoint(index++, 1, 5, 0));//Points.Add(new CPoint(index++, 1, 6, 0));//Points.Add(new CPoint(index++, 2, 1, 0));//Points.Add(new CPoint(index++, 2, 2, 0));Points.Add(new CPoint(index++, 2, 3, 0));Points.Add(new CPoint(index++, 2, 4, 0));//Points.Add(new CPoint(index++, 2, 5, 0));//Points.Add(new CPoint(index++, 2, 6, 0));Points.Add(new CPoint(index++, 3, 1, 0));Points.Add(new CPoint(index++, 3, 2, 0));Points.Add(new CPoint(index++, 3, 3, 0));Points.Add(new CPoint(index++, 3, 4, 0));Points.Add(new CPoint(index++, 3, 5, 0));Points.Add(new CPoint(index++, 3, 6, 0));Points.Add(new CPoint(index++, 4, 1, 0));Points.Add(new CPoint(index++, 4, 2, 0));Points.Add(new CPoint(index++, 4, 3, 0));Points.Add(new CPoint(index++, 4, 4, 0));Points.Add(new CPoint(index++, 4, 5, 0));Points.Add(new CPoint(index++, 4, 6, 0));//Points.Add(new CPoint(index++, 5, 1, 0));//Points.Add(new CPoint(index++, 5, 2, 0));Points.Add(new CPoint(index++, 5, 3, 0));Points.Add(new CPoint(index++, 5, 4, 0));//Points.Add(new CPoint(index++, 5, 5, 0));//Points.Add(new CPoint(index++, 5, 6, 0));//Points.Add(new CPoint(index++, 6, 1, 0));//Points.Add(new CPoint(index++, 6, 2, 0));Points.Add(new CPoint(index++, 6, 3, 0));Points.Add(new CPoint(index++, 6, 4, 0));//Points.Add(new CPoint(index++, 6, 5, 0));//Points.Add(new CPoint(index++, 6, 6, 0));Squres = CSqure.GetAllSqures(Points);}private int drawIndex = 0;private void Timer_Tick(object sender, EventArgs e){drawIndex++;if(drawIndex>Squres.Count()){drawIndex = 1;}Invalidate();}protected override void OnPaint(PaintEventArgs e){e.Graphics.SmoothingMode = System.Drawing.Drawing2D.SmoothingMode.AntiAlias;foreach (var item in Points){e.Graphics.FillEllipse(SystemBrushes.GrayText, item.DrawCircle()) ;}return;int indexr = 0;int indexg = 0;int indexb = 0;int start = 127;int r = start;int g = start;int b = start;int step = 40;int index=0;foreach (var item in Squres){index++;indexr += step;if(index != drawIndex){continue;}if(indexr>255- start){indexg += step;if (indexg > 255 - start){indexb += step;if (indexb > 255 - start){indexb = 0;indexg = 0;indexb = 0;}indexg = 0;}indexr = 0;}var color = Color.FromArgb(255, r + indexr, g + indexg, b + indexb);var color2 = Color.FromArgb(255,255- r - indexr, 255 - g - indexg, 255 - b - indexb);using (Pen p = new Pen(color))using (Brush b2 = new SolidBrush(color))using (Brush b3 = new SolidBrush(color2)){e.Graphics.FillPolygon(b2, item.DrawSqure());e.Graphics.DrawPolygon(p, item.DrawSqure());var size = e.Graphics.MeasureString($"{index}",this.Font);e.Graphics.DrawString($"{index}",this.Font, b3, item.DrawCenter(size));e.Graphics.DrawString($"{index}.边长：{item.C.ToString("0.00")}", this.Font, b3, new PointF(50,53));}}base.OnPaint(e);}}
}