Maximum and Minimum number of regions inside a circle divided by n lines

What is the maximum number of regions within a circle that can be formed by n distinct lines intersecting inside the circle? Express your answer in fraction form. (Use LaTeX syntax, enclose your answer in dollar signs ($): $\frac{1}{2}$)

Answer:

Correct! The answer is $\binom n2+\binom n1 + \binom n0=\binom {n+1}{2} + 1=\frac{n^{2}+n+2}{2}$

Incorrect! Try again.

What is the maximum number of regions within a circle that can be formed by n distinct lines intersecting outside the circle?
Answer:

Correct! The answer is $n+1$

Incorrect! Try again.

What is the minimum number of regions within a circle that can be formed by n distinct lines crossing the circle?
Answer:

Correct! The answer is $n+1$

Incorrect! Try again.

(Based on Mathcounts National 1990 Target Problem 8 and Mathcounts Chapter 1992 Sprint problem 5)
import simplegui import random import math WIDTH=600 HEIGHT=250 CIRCLE_CENTER=(150,120) CIRCLE_RIGHT_CENTER=(150+300,120) CIRCLE_RADIUS=95 messageNumLinesPos=[120,20] messageNumRegionsPos=[120,240] YES_BUTTON_POS_BOTTOM_LEFT=[450,60] YES_BUTTON_POS_BOTTOM_RIGHT=[550,60] YES_BUTTON_POS_TOP_RIGHT=[550,30] YES_BUTTON_POS_TOP_LEFT=[450,30] YES_BUTTON_COORDS=[YES_BUTTON_POS_BOTTOM_LEFT,YES_BUTTON_POS_BOTTOM_RIGHT,YES_BUTTON_POS_TOP_RIGHT,YES_BUTTON_POS_TOP_LEFT] NO_BUTTON_POS_BOTTOM_LEFT=[450,110] NO_BUTTON_POS_BOTTOM_RIGHT=[550,110] NO_BUTTON_POS_TOP_RIGHT=[550,80] NO_BUTTON_POS_TOP_LEFT=[450,80] NO_BUTTON_COORDS=[NO_BUTTON_POS_BOTTOM_LEFT,NO_BUTTON_POS_BOTTOM_RIGHT,NO_BUTTON_POS_TOP_RIGHT,NO_BUTTON_POS_TOP_LEFT] NEXT_BUTTON_POS_BOTTOM_LEFT=[450,250] NEXT_BUTTON_POS_BOTTOM_RIGHT=[550,250] NEXT_BUTTON_POS_TOP_RIGHT=[550,220] NEXT_BUTTON_POS_TOP_LEFT=[450,220] NEXT_BUTTON_COORDS=[NEXT_BUTTON_POS_BOTTOM_LEFT,NEXT_BUTTON_POS_BOTTOM_RIGHT,NEXT_BUTTON_POS_TOP_RIGHT,NEXT_BUTTON_POS_TOP_LEFT] clickedYes = False clickedNo = False scrollup = 0 countScrolls=0 points = [(20,200),(260,80),(20,20),(260,200),(60,240),(270,40)] pointsRight = [(20+300,200),(260+300,80),(20+300,20),(260+300,200),(155+300,145)] pointsIds = [(20,72),(60,35),(140,95),(254,30),(155,158)] messageNumLines="n=1" messageNumRegions = "2 regions" n=1 # Handler for buttons def increasen(): global messageNumLines ,messageNumRegions,n if (n<5): n +=1 messageNumLines = "n=" + str(n) messageNumRegions = str((n**2+n+2)/2) + " regions" def decreasen(): global messageNumLines ,messageNumRegions,n if (n>1): n -=1 messageNumLines = "n=" + str(n) messageNumRegions = str((n**2+n+2)/2) + " regions" def handleCase1(canvas): canvas.draw_line(points[0],points[1],3,"Yellow") m = (points[0][1]-points[1][1])*1.0/(points[0][0]-points[1][0]) b = points[0][1]-m*points[0][0] for i in range(1,2600): x = random.randrange(1,300) y = random.randrange(1,250) if (x-150)**2+(y-120)**2<95**2: if (y>m*x+b): canvas.draw_circle((x,y),1,2,"Red") else: canvas.draw_circle((x,y),1,2,"White") def handleCase2(canvas): global messageNumLinesPos,messageNumRegionsPos canvas.draw_line(points[0],points[1],3,"Yellow") canvas.draw_line(points[2],points[3],3,"Red") messageNumLinesPos=[120,20] messageNumRegionsPos=[120,240] #find coefficients of the 2 lines m1 = (points[0][1]-points[1][1])*1.0/(points[0][0]-points[1][0]) b1 = points[0][1]-m1*points[0][0] m2 = (points[2][1]-points[3][1])*1.0/(points[2][0]-points[3][0]) b2 = points[2][1]-m2*points[2][0] for i in range(1,2600): x = random.randrange(1,300) y = random.randrange(1,250) if (x-150)**2+(y-120)**2<95**2: if (y>m1*x+b1) and (y>m2*x+b2): canvas.draw_circle((x,y),1,2,"Red") elif (y>m1*x+b1) and (y<m2*x+b2): canvas.draw_circle((x,y),1,2,"White") elif (y<m1*x+b1) and (y<m2*x+b2): canvas.draw_circle((x,y),1,2,"Cyan") else: canvas.draw_circle((x,y),1,2,"Green") def drawYesNo(canvas,color): canvas.draw_polygon(YES_BUTTON_COORDS,12,color,color) canvas.draw_text("Yes", [YES_BUTTON_POS_BOTTOM_LEFT[0]+(YES_BUTTON_POS_BOTTOM_RIGHT[0]-YES_BUTTON_POS_BOTTOM_LEFT[0])/4,YES_BUTTON_POS_TOP_LEFT[1]+(YES_BUTTON_POS_BOTTOM_LEFT[1]-YES_BUTTON_POS_TOP_LEFT[1])/2],18,"Black") canvas.draw_polygon(NO_BUTTON_COORDS,12,color,color) canvas.draw_text("No", [NO_BUTTON_POS_BOTTOM_LEFT[0]+(NO_BUTTON_POS_BOTTOM_RIGHT[0]-NO_BUTTON_POS_BOTTOM_LEFT[0])/4,NO_BUTTON_POS_TOP_LEFT[1]+(NO_BUTTON_POS_BOTTOM_LEFT[1]-NO_BUTTON_POS_TOP_LEFT[1])/2],18,"Black") def handleCase3a(canvas): canvas.draw_line(points[0],points[1],3,"Yellow") canvas.draw_line(points[2],points[3],3,"Red") #find coefficients of the 2 lines m1 = (points[0][1]-points[1][1])*1.0/(points[0][0]-points[1][0]) b1 = points[0][1]-m1*points[0][0] m2 = (points[2][1]-points[3][1])*1.0/(points[2][0]-points[3][0]) b2 = points[2][1]-m2*points[2][0] #intersection point xint = (b1-b2)*1.0/(m2-m1) yint = m1*xint + b1 #find 3rd line with common point x3=xint+130 y3=10*yint/xint*x3 m3=-(yint-y3)/130.0 b3=y3-m3*x3 canvas.draw_line([xint,yint],[xint+130,10*yint*1.0/xint*(xint+130)],3,"Cyan") canvas.draw_line([xint,yint],[xint-130,250-10*yint*1.0/xint*(xint+130)],3,"Cyan") for i in range(1,2600): x = random.randrange(1,300) y = random.randrange(1,250) if (x-150)**2+(y-120)**2<95**2: if (y>m1*x+b1) and (y>m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"Red") elif (y>m1*x+b1) and (y>m2*x+b2) and (y>m3*x+b3): canvas.draw_circle((x,y),1,2,"Cyan") elif (y>m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"White") elif (y<m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"Green") elif (y<m1*x+b1) and (y<m2*x+b2) and (y>m3*x+b3): canvas.draw_circle((x,y),1,2,"Yellow") else: canvas.draw_circle((x,y),1,2,"Fuchsia") global messageNumRegions,messageNumRegionsPos messageNumRegions = "6 regions. Can we do better?" if (messageNumLinesPos[0]>70): messageNumLinesPos[0]-=10 if (messageNumRegionsPos[0]>10): messageNumRegionsPos[0] -= 10 drawYesNo(canvas,"orange") def handleCase3(canvas): #redraw left side handleCase3a(canvas) #delete yes/no buttons drawYesNo(canvas,"black") canvas.draw_circle(CIRCLE_RIGHT_CENTER,CIRCLE_RADIUS,3,"Green") canvas.draw_line(pointsRight[0],pointsRight[1],3,"Yellow") canvas.draw_line(pointsRight[2],pointsRight[3],3,"Red") #find coefficients of the 2 lines m1 = (pointsRight[0][1]-pointsRight[1][1])*1.0/(pointsRight[0][0]-pointsRight[1][0]) b1 = pointsRight[0][1]-m1*pointsRight[0][0] m2 = (pointsRight[2][1]-pointsRight[3][1])*1.0/(pointsRight[2][0]-pointsRight[3][0]) b2 = pointsRight[2][1]-m2*pointsRight[2][0] #intersection point xint = (b1-b2)*1.0/(m2-m1) yint = m1*xint + b1 canvas.draw_line([xint-50,yint],[xint+130-50,10*yint*1.0/xint*(xint+130-50)],3,"Cyan") canvas.draw_line([xint-50,yint],[xint-130-50,250-10*yint*1.0/xint*(xint+130-50)],3,"Cyan") #find 3rd line params x3=xint+130-50 y3=10*yint/xint*x3 m3=-(yint-y3)/130.0 b3=y3-m3*x3 for i in range(1,2600): x = random.randrange(300,600) y = random.randrange(1,250) if (x-150-300)**2+(y-120)**2<95**2: if (y>m1*x+b1) and (y>m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"Red") elif (y>m1*x+b1) and (y>m2*x+b2) and (y>m3*x+b3): canvas.draw_circle((x,y),1,2,"Cyan") elif (y>m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"White") elif (y<m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"Green") elif (y<m1*x+b1) and (y<m2*x+b2) and (y>m3*x+b3): canvas.draw_circle((x,y),1,2,"Yellow") elif (y<m1*x+b1) and (y>m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"Orange") else: canvas.draw_circle((x,y),1,2,"Fuchsia") canvas.draw_text("Yes! 7 regions", [messageNumRegionsPos[0]+420,messageNumRegionsPos[1]], 28, "Red") def handleCase4(canvas): canvas.draw_circle(CIRCLE_CENTER,CIRCLE_RADIUS,3,"Green") canvas.draw_line(points[0],points[1],3,"Yellow") canvas.draw_line(points[2],points[3],3,"Red") #find coefficients of the 2 lines m1 = (points[0][1]-points[1][1])*1.0/(points[0][0]-points[1][0]) b1 = points[0][1]-m1*points[0][0] m2 = (points[2][1]-points[3][1])*1.0/(points[2][0]-points[3][0]) b2 = points[2][1]-m2*points[2][0] #intersection point xint = (b1-b2)*1.0/(m2-m1) yint = m1*xint + b1 canvas.draw_line([xint-50,yint],[xint+130-50,10*yint*1.0/xint*(xint+130-50)],3,"Cyan") canvas.draw_line([xint-50,yint],[xint-130-50,250-10*yint*1.0/xint*(xint+130-50)],3,"Cyan") #find 3rd line params x3=xint+130-50 y3=10*yint/xint*x3 m3=-(yint-y3)/130.0 b3=y3-m3*x3 #4th line canvas.draw_line(points[4],points[5],3,"Brown") m4 = (points[4][1]-points[5][1])*1.0/(points[4][0]-points[5][0]) b4 = points[4][1]-m4*points[4][0] for i in range(1,2600): x = random.randrange(1,300) y = random.randrange(1,250) if (x-150)**2+(y-120)**2<95**2: if (y>m1*x+b1) and (y>m2*x+b2) and (y<m3*x+b3) and (y<m4*x+b4): canvas.draw_circle((x,y),1,2,"Red") elif (y>m1*x+b1) and (y>m2*x+b2) and (y<m3*x+b3) and (y>m4*x+b4): canvas.draw_circle((x,y),1,2,"Grey") elif (y>m1*x+b1) and (y>m2*x+b2) and (y>m3*x+b3) and (y<m4*x+b4): canvas.draw_circle((x,y),1,2,"Cyan") elif (y>m1*x+b1) and (y>m2*x+b2) and (y>m3*x+b3) and (y>m4*x+b4): canvas.draw_circle((x,y),1,2,"Blue") elif (y>m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3) and (y<m4*x+b4): canvas.draw_circle((x,y),1,2,"White") elif (y>m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3) and (y>m4*x+b4): canvas.draw_circle((x,y),1,2,"Purple") elif (y<m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3) and (y<m4*x+b4): canvas.draw_circle((x,y),1,2,"Green") elif (y<m1*x+b1) and (y<m2*x+b2) and (y>m3*x+b3): canvas.draw_circle((x,y),1,2,"Yellow") elif (y<m1*x+b1) and (y>m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"Orange") else: canvas.draw_circle((x,y),1,2,"Fuchsia") global messageNumLinesPos,messageNumRegionsPos messageNumLinesPos=[130,20] messageNumRegionsPos=[130,240] canvas.draw_text("When we add a line, ", [300,20-scrollup], 28, "Red") canvas.draw_text("the new line should never ", [300,48-scrollup], 28, "Red") canvas.draw_text("go through an existing ", [300,76-scrollup], 28, "Red") canvas.draw_text("intersection point. ", [300,104-scrollup], 28, "Red") canvas.draw_text("If it does, it will only add",[300,132-scrollup],28,"Red") canvas.draw_text("2 news regions. ",[300,160-scrollup],28,"Red") canvas.draw_text("Instead the new line should ",[300,188-scrollup],28,"Green") canvas.draw_text("intersect every previous ",[300,216-scrollup],28,"Green") canvas.draw_text("line at a point where no",[300,244-scrollup],28,"Green") canvas.draw_text("two lines have yet ",[300,272-scrollup],28,"Green") canvas.draw_text("intersected.",[300,300-scrollup],28,"Green") canvas.draw_text("Thus the nth line will have",[300,328-scrollup],28,"Green") canvas.draw_text("n-1 intersections. Each of ",[300,356-scrollup],28,"Green") canvas.draw_text("these split an existing ",[300,382-scrollup],28,"Green") canvas.draw_text("region into 2, therefore ",[300,410-scrollup],28,"Green") canvas.draw_text("creating n new regions.",[300,438-scrollup],28,"Green") def handleCases(canvas): if n==1: handleCase1(canvas) elif n==2: handleCase2(canvas) elif n==3 and not clickedYes and not clickedNo: handleCase3a(canvas) elif n==3 and (clickedYes or clickedNo): handleCase3(canvas) elif n==4: handleCase4(canvas) def mclick(pos): global clickedYes,clickedNo #Yes button click if n==3 and pos[0]>YES_BUTTON_POS_BOTTOM_LEFT[0] and pos[0]<YES_BUTTON_POS_BOTTOM_RIGHT[0] and pos[1]>YES_BUTTON_POS_TOP_LEFT[1] and pos[1]<YES_BUTTON_POS_BOTTOM_LEFT[1]: clickedYes=True #No button click elif n==3 and pos[0]>NO_BUTTON_POS_BOTTOM_LEFT[0] and pos[0]<NO_BUTTON_POS_BOTTOM_RIGHT[0] and pos[1]>NO_BUTTON_POS_TOP_LEFT[1] and pos[1]<NO_BUTTON_POS_BOTTOM_LEFT[1]: clickedNo = True #Next button elif n==5 and pos[0]>NEXT_BUTTON_POS_BOTTOM_LEFT[0] and pos[0]<NEXT_BUTTON_POS_BOTTOM_RIGHT[0] and pos[1]>NEXT_BUTTON_POS_TOP_LEFT[1] and pos[1]<NEXT_BUTTON_POS_BOTTOM_LEFT[1]: clickedYes = True def timer_handler(): global scrollup, countScrolls if n==4 and countScrolls<18: scrollup +=28 countScrolls +=1 elif n==4 and countScrolls==18: countScrolls=0 scrollup=0 scrollup -= 28*9 else: countScrolls=0 scrollup=0 def sweepLine(canvas): pass def draw(canvas): if n>4: messageNumLines1 = "For n lines, the maximum number of " messageNumLines2 = "regions is therefore given by the sum " messageNumLines3 = "2+2+3+4+5+...+n-1+n. " messageNumLines4 = "Can you figure how what does this" messageNumLines5 = "sum add up to?" messageNumLines6 = "Alternatively imagine we divide the " messageNumLines7 = "circle using n lines into the max number" messageNumLines8 = "of regions. Imagine a line that lies " messageNumLines9 = "initially outside the circle. Because" messageNumLines10 = "this line intersects the n lines at " messageNumLines11 = "n points it intersects n+1 regions. " messageNumLines12 = "Now move this line across all the " messageNumLines13 = "intersections of the n lines (inside" messageNumLines14 = "the circle). Since each of the n lines" messageNumLines15 = "intersects every other line there are" messageNumLines16 = "n choose 2 intersections. Each time " messageNumLines17 = "the sweeping line crosses one of these" messageNumLines18 = "intersections in enters a new region." messageNumLines19 = "Therefore there are (n choose 2) + " messageNumLines20 = "(n choose 1) + (n choose 0) maximum " messageNumLines21 = "regions. Interestingly this is equal" messageNumLines22 = "to (n+1) choose 2 +1. Why?" canvas.draw_text(messageNumLines1,[10,20],18,"Red") canvas.draw_text(messageNumLines2,[10,40],18,"Red") canvas.draw_text(messageNumLines3,[10,60],18,"Red") canvas.draw_text(messageNumLines4,[10,80],18,"Red") canvas.draw_text(messageNumLines5,[10,100],18,"Red") canvas.draw_text(messageNumLines6,[10,120],18,"Green") canvas.draw_text(messageNumLines7,[10,140],18,"Green") canvas.draw_text(messageNumLines8,[10,160],18,"Green") canvas.draw_text(messageNumLines9,[10,180],18,"Green") canvas.draw_text(messageNumLines10,[10,200],18,"Green") canvas.draw_text(messageNumLines11,[10,220],18,"Green") canvas.draw_text(messageNumLines12,[10,240],18,"Green") canvas.draw_text(messageNumLines13,[300,20],18,"Green") canvas.draw_text(messageNumLines14,[300,40],18,"Green") canvas.draw_text(messageNumLines15,[300,60],18,"Green") canvas.draw_text(messageNumLines16,[300,80],18,"Green") canvas.draw_text(messageNumLines17,[300,100],18,"Green") canvas.draw_text(messageNumLines18,[300,120],18,"Green") canvas.draw_text(messageNumLines19,[300,140],18,"Green") canvas.draw_text(messageNumLines20,[300,160],18,"Green") canvas.draw_text(messageNumLines21,[300,180],18,"Green") canvas.draw_text(messageNumLines22,[300,200],18,"Green") #canvas.draw_polygon(NEXT_BUTTON_COORDS,12,"orange","orange") #canvas.draw_text("Next",[NEXT_BUTTON_POS_BOTTOM_LEFT[0]+(NEXT_BUTTON_POS_BOTTOM_RIGHT[0]-NEXT_BUTTON_POS_BOTTOM_LEFT[0])/4,NEXT_BUTTON_POS_TOP_LEFT[1]+(NEXT_BUTTON_POS_BOTTOM_LEFT[1]-NEXT_BUTTON_POS_TOP_LEFT[1])/2],18,"Black") #if clickedYes: # sweepLine(canvas) else: canvas.draw_text(messageNumLines,messageNumLinesPos, 28, "Red") canvas.draw_circle(CIRCLE_CENTER,CIRCLE_RADIUS,3,"Green") handleCases(canvas) canvas.draw_text(messageNumRegions, messageNumRegionsPos, 28, "Red") # Create a frame and assign callbacks to event handlers frame = simplegui.create_frame("How many regions?", WIDTH, HEIGHT) frame.add_button("From n=1 to n=4", increasen) frame.add_button("From n=4 to n=1", decreasen) frame.set_mouseclick_handler(mclick) frame.set_draw_handler(draw) timer = simplegui.create_timer(1000,timer_handler) timer.start() # Start the frame animation frame.start()
import simplegui import random import math WIDTH=600 HEIGHT=250 CIRCLE_CENTER=(150,120) CIRCLE_RIGHT_CENTER=(150+300,120) CIRCLE_RADIUS=95 messageNumLinesPos=[120,20] messageNumRegionsPos=[120,240] messageNumLines="n=1" messageNumRegions = "2 regions" n=1 points = [(20,220),(320,80),(20,80),(320,110),(20,200),(320,200),(20,20),(320,100)] # Handler for buttons def increasen(): global messageNumLines ,messageNumRegions,n if (n<5): n +=1 messageNumLines = "n=" + str(n) messageNumRegions = str((n+1)) + " regions" def decreasen(): global messageNumLines ,messageNumRegions,n if (n>1): n -=1 messageNumLines = "n=" + str(n) messageNumRegions = str((n+1)) + " regions" def handleCase1(canvas): canvas.draw_line(points[0],points[1],3,"Yellow") m = (points[0][1]-points[1][1])*1.0/(points[0][0]-points[1][0]) b = points[0][1]-m*points[0][0] for i in range(1,600): x = random.randrange(1,300) y = random.randrange(1,250) if (x-150)**2+(y-120)**2<95**2: if (y>m*x+b): canvas.draw_circle((x,y),1,2,"Red") else: canvas.draw_circle((x,y),1,2,"White") def handleCase2(canvas): global messageNumLinesPos,messageNumRegionsPos canvas.draw_line(points[0],points[1],3,"Yellow") canvas.draw_line(points[2],points[3],3,"Red") messageNumLinesPos=[120,20] messageNumRegionsPos=[120,240] #find coefficients of the 2 lines m1 = (points[0][1]-points[1][1])*1.0/(points[0][0]-points[1][0]) b1 = points[0][1]-m1*points[0][0] m2 = (points[2][1]-points[3][1])*1.0/(points[2][0]-points[3][0]) b2 = points[2][1]-m2*points[2][0] for i in range(1,600): x = random.randrange(1,300) y = random.randrange(1,250) if (x-150)**2+(y-120)**2<95**2: if (y>m1*x+b1) and (y>m2*x+b2): canvas.draw_circle((x,y),1,2,"Red") elif (y>m1*x+b1) and (y<m2*x+b2): canvas.draw_circle((x,y),1,2,"White") elif (y<m1*x+b1) and (y<m2*x+b2): canvas.draw_circle((x,y),1,2,"Cyan") else: canvas.draw_circle((x,y),1,2,"Green") def handleCase3(canvas): canvas.draw_line(points[0],points[1],3,"Yellow") canvas.draw_line(points[2],points[3],3,"Red") canvas.draw_line(points[4],points[5],3,"Cyan") #find coefficients of the 3 lines m1 = (points[0][1]-points[1][1])*1.0/(points[0][0]-points[1][0]) b1 = points[0][1]-m1*points[0][0] m2 = (points[2][1]-points[3][1])*1.0/(points[2][0]-points[3][0]) b2 = points[2][1]-m2*points[2][0] m3 = (points[4][1]-points[5][1])*1.0/(points[4][0]-points[5][0]) b3 = points[4][1]-m2*points[4][0] for i in range(1,600): x = random.randrange(1,300) y = random.randrange(1,250) if (x-150)**2+(y-120)**2<95**2: if (y>m1*x+b1) and (y>m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"Red") elif (y>m1*x+b1) and (y>m2*x+b2) and (y>m3*x+b3): canvas.draw_circle((x,y),1,2,"Cyan") elif (y>m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"White") elif (y<m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"Green") elif (y<m1*x+b1) and (y<m2*x+b2) and (y>m3*x+b3): canvas.draw_circle((x,y),1,2,"Yellow") else: canvas.draw_circle((x,y),1,2,"Fuchsia") def handleCase4(canvas): canvas.draw_line(points[0],points[1],3,"Yellow") canvas.draw_line(points[2],points[3],3,"Red") canvas.draw_line(points[4],points[5],3,"Cyan") canvas.draw_line(points[6],points[7],3,"Brown") #find coefficients of the 4 lines m1 = (points[0][1]-points[1][1])*1.0/(points[0][0]-points[1][0]) b1 = points[0][1]-m1*points[0][0] m2 = (points[2][1]-points[3][1])*1.0/(points[2][0]-points[3][0]) b2 = points[2][1]-m2*points[2][0] m3 = (points[4][1]-points[5][1])*1.0/(points[4][0]-points[5][0]) b3 = points[4][1]-m2*points[4][0] m4 = (points[6][1]-points[7][1])*1.0/(points[6][0]-points[7][0]) b4 = points[6][1]-m2*points[6][0] for i in range(1,600): x = random.randrange(1,300) y = random.randrange(1,250) if (x-150)**2+(y-120)**2<95**2: if (y>m1*x+b1) and (y>m2*x+b2) and (y<m3*x+b3) and (y<m4*x+b4): canvas.draw_circle((x,y),1,2,"Red") elif (y>m1*x+b1) and (y>m2*x+b2) and (y<m3*x+b3) and (y>m4*x+b4): canvas.draw_circle((x,y),1,2,"Grey") elif (y>m1*x+b1) and (y>m2*x+b2) and (y>m3*x+b3) and (y<m4*x+b4): canvas.draw_circle((x,y),1,2,"Cyan") elif (y>m1*x+b1) and (y>m2*x+b2) and (y>m3*x+b3) and (y>m4*x+b4): canvas.draw_circle((x,y),1,2,"Blue") elif (y>m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3) and (y<m4*x+b4): canvas.draw_circle((x,y),1,2,"White") elif (y>m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3) and (y>m4*x+b4): canvas.draw_circle((x,y),1,2,"Purple") elif (y<m1*x+b1) and (y<m2*x+b2) and (y<m3*x+b3) and (y<m4*x+b4): canvas.draw_circle((x,y),1,2,"Green") elif (y<m1*x+b1) and (y<m2*x+b2) and (y>m3*x+b3): canvas.draw_circle((x,y),1,2,"Yellow") elif (y<m1*x+b1) and (y>m2*x+b2) and (y<m3*x+b3): canvas.draw_circle((x,y),1,2,"Orange") else: canvas.draw_circle((x,y),1,2,"Fuchsia") def handleCases(canvas): if n==1: handleCase1(canvas) elif n==2: handleCase2(canvas) elif n==3: handleCase3(canvas) elif n==4: handleCase4(canvas) def draw(canvas): if n>4: messageNumLines1 = "Notice that every time we add one extra" messageNumLines2 = "line, the best we can do is cut one " messageNumLines3 = "existing region into 2. We can not create" messageNumLines4 = "more than one new region because that" messageNumLines5 = "would involve the new line intersecting " messageNumLines6 = "an existing line inside the circle. " messageNumLines7 = "But all intersections between the lines " messageNumLines8 = "occur outside the circle." messageNumLines9 = "Therefore the maximum number of regions" messageNumLines10 = "we can create with n lines intersecting" messageNumLines11 = "outside the circle is n+1." canvas.draw_text(messageNumLines1,[10,20],18,"Green") canvas.draw_text(messageNumLines2,[10,40],18,"Green") canvas.draw_text(messageNumLines3,[10,60],18,"Green") canvas.draw_text(messageNumLines4,[10,80],18,"Green") canvas.draw_text(messageNumLines5,[10,100],18,"Green") canvas.draw_text(messageNumLines6,[10,120],18,"Green") canvas.draw_text(messageNumLines7,[10,140],18,"Green") canvas.draw_text(messageNumLines8,[10,160],18,"Green") canvas.draw_text(messageNumLines9,[10,180],18,"Green") canvas.draw_text(messageNumLines10,[10,200],18,"Green") canvas.draw_text(messageNumLines11,[10,220],18,"Green") else: canvas.draw_text(messageNumLines,messageNumLinesPos, 28, "Red") canvas.draw_circle(CIRCLE_CENTER,CIRCLE_RADIUS,3,"Green") handleCases(canvas) canvas.draw_text(messageNumRegions, messageNumRegionsPos, 28, "Red") # Create a frame and assign callbacks to event handlers frame = simplegui.create_frame("How many regions?", WIDTH, HEIGHT) frame.add_button("From n=1 to n=4", increasen) frame.add_button("From n=4 to n=1", decreasen) frame.set_draw_handler(draw) # Start the frame animation frame.start()
import simplegui import random import math WIDTH=600 HEIGHT=250 def draw(canvas): messageNumLines1 = "The minimum number of regions we" messageNumLines2 = "can divide the circle by n lines " messageNumLines3 = "will occur when none of the n lines" messageNumLines4 = "intersect inside the circle." messageNumLines5 = "If the n lines are parallel " messageNumLines6 = "they will create n+1 regions inside the " messageNumLines7 = "circle. If a few intersect outside the " messageNumLines8 = "circle will that change the number of " messageNumLines9 = "regions they divide the circle?" canvas.draw_text(messageNumLines1,[10,20],18,"Green") canvas.draw_text(messageNumLines2,[10,40],18,"Green") canvas.draw_text(messageNumLines3,[10,60],18,"Green") canvas.draw_text(messageNumLines4,[10,80],18,"Green") canvas.draw_text(messageNumLines5,[10,100],18,"Green") canvas.draw_text(messageNumLines6,[10,120],18,"Green") canvas.draw_text(messageNumLines7,[10,140],18,"Green") canvas.draw_text(messageNumLines8,[10,160],18,"Green") canvas.draw_text(messageNumLines9,[10,180],18,"Green") # Create a frame and assign callbacks to event handlers frame = simplegui.create_frame("How many regions?", WIDTH, HEIGHT) frame.set_draw_handler(draw) # Start the frame animation frame.start()