One-circle-in-a-plane-can-produce-one-closed-region-at-most-It-produces-one-closed-region-at-least-Two-circles-in-a-plane-can-produce-at-most-three-regions-They-produce-at-least-two-regions

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Question Number 4033 by Rasheed Soomro last updated on 27/Dec/15

$$\mathrm{One}\mathrm{circle}\mathrm{in}\mathrm{a}\mathrm{plane}\mathrm{can}\mathrm{produce}\mathrm{one}\mathrm{closed}$$$$\mathrm{region}\mathrm{at}\mathrm{most}(\mathrm{It}\mathrm{produces}\mathrm{one}\mathrm{closed}\mathrm{region}$$$$\mathrm{at}\mathrm{least}).\mathrm{Two}\mathrm{circles}\mathrm{in}\mathrm{a}\mathrm{plane}\mathrm{can}\mathrm{produce}$$$$\mathrm{at}\mathrm{most}\mathrm{three}\mathrm{regions}(\mathrm{They}\mathrm{produce}\mathrm{at}\mathrm{least}$$$$\mathrm{two}\mathrm{regions}).\mathrm{Three}\mathrm{circles}\mathrm{can}\mathrm{produce}\mathrm{seven}$$$$\mathrm{closed}\mathrm{regions}\mathrm{at}\mathrm{most}(\mathrm{They}\mathrm{produce}\mathrm{three}$$$$\mathrm{closed}\mathrm{regions}\mathrm{at}\mathrm{least}).$$$$$$$$\mathrm{How}\mathrm{many}\mathbf{distinct}\mathrm{closed}\mathrm{regions}\mathrm{could}\mathrm{be}\mathrm{produced}$$$$\mathrm{by}4,5\mathrm{and}\mathrm{n}\mathrm{circles}\mathrm{at}\mathrm{most}?$$$$\mathrm{If}\mathrm{n}\mathrm{circles}\mathrm{can}\mathrm{produce}\mathrm{f}\left(\mathrm{n}\right)\mathrm{closed}\mathrm{regions}$$$$\mathrm{at}\mathrm{most}(\mathrm{Of}\mathrm{course}\mathrm{they}\mathrm{produce}\mathrm{n}\mathrm{closed}$$$$\mathrm{regions}\mathrm{at}\mathrm{least}),\mathrm{can}\mathrm{they}\mathrm{produce}\mathrm{any}$$$$\mathrm{number}\mathrm{of}\mathrm{closed}\mathrm{regions}\mathrm{between}\mathrm{n}\mathrm{and}\mathrm{f}\left(\mathrm{n}\right)?$$

Commented by Rasheed Soomro last updated on 27/Dec/15

$$\mathrm{For}\mathrm{three}\mathrm{circles}3,4,5,6\mathrm{and}7\mathrm{closed}\mathrm{regions}$$$$\mathrm{can}\mathrm{be}\mathrm{made}.$$

Commented by Rasheed Soomro last updated on 27/Dec/15

$$\mathrm{Which}\mathrm{area}\mathrm{of}\mathrm{mathematics}\mathrm{does}\mathrm{this}\mathrm{problem}$$$$\mathrm{belong}\mathrm{to}?$$

Commented by Yozzii last updated on 27/Dec/15

Commented by Yozzii last updated on 27/Dec/15

$$Ionlyfirstmetthisideainsettheory$$$$butf\left(n\right)=2^n-1isvalidonlyif0⩽n⩽3.$$$$4circlesappeartogive13closed$$$$regionsatmost.$$

Commented by Yozzii last updated on 27/Dec/15

Commented by Yozzii last updated on 27/Dec/15

$$5circlesappeartoproduceatmost$$$$21closedregions.$$$$n=0:f\left(0\right)=0$$$$n=1:f\left(1\right)=1$$$$n=2:f\left(2\right)=3$$$$n=3:f\left(3\right)=7$$$$n=4:f\left(4\right)=13$$$$n=5:f\left(5\right)=21$$$$n=6:f\left(6\right)=31$$$$f\left(1\right)-f\left(0\right)=1=1\times 2^0$$$$f\left(2\right)-f\left(1\right)=2=1\times 2$$$$f\left(3\right)-f\left(2\right)=4=2\times 2$$$$f\left(4\right)-f\left(3\right)=6=3\times 2$$$$f\left(5\right)-f\left(4\right)=8=4\times 2$$$$f\left(6\right)-f\left(5\right)=10=5\times 2$$$$n⩾1,f(n+1)-f\left(n\right)=2n,f\left(1\right)=1$$$$f(n+1)=f\left(n\right)+2n$$$$f(n+1)-f\left(n\right)=2n$$$$f\left(n\right)-f(n-1)=2(n-1)$$$$f(n-1)-f(n-2)=2(n-2)$$$$⋮$$$$f\left(4\right)-f\left(3\right)=2\times 3$$$$f\left(3\right)-f\left(2\right)=2\times 2$$$$f\left(2\right)-f\left(1\right)=2\times 1$$$$\Rightarrow f(n+1)-f\left(1\right)=2\underset{r=1}{\sum ^n}r$$$$f(n+1)=f\left(1\right)+2\times 0.5n(n+1)$$$$f(n+1)=1+n(n+1)$$$$\Rightarrow f\left(n\right)=1+n(n-1)$$$$f\left(n\right)=n^2-n+1n⩾1$$$$$$

Commented by prakash jain last updated on 27/Dec/15

$$n-circlewillgive\mathrm{maximum}{2}^n-1\mathrm{closed}\mathrm{region}.$$$${}^n{C}_1{+}^n{C}_2+..+{}^n{C}_n=2^n-1$$$$\mathrm{Each}\mathrm{term}\mathrm{on}\mathrm{LHS}{}^n{C}_r\mathrm{is}\mathrm{created}\mathrm{by}\mathrm{number}$$$$\mathrm{of}\mathrm{ways}r\mathrm{circles}\mathrm{can}\mathrm{intersect}.\mathrm{Each}\mathrm{of}\mathrm{those}$$$$\mathrm{intersection}\mathrm{form}\mathrm{a}\mathrm{closed}\mathrm{area}.$$$$\mathrm{Since}\mathrm{all}\mathrm{circles}\mathrm{can}\mathrm{be}\mathrm{independently}\mathrm{drawn}$$$$\mathrm{you}\mathrm{can}\mathrm{create}\mathrm{any}\mathrm{number}\mathrm{of}\mathrm{closed}\mathrm{regions}$$$$\mathrm{from}n\mathrm{to}{2}^n-1.$$

Commented by Rasheed Soomro last updated on 27/Dec/15

$$\mathrm{But}\mathrm{sir}{\mathrm{Yozzii}}^{\prime }\mathrm{s}\mathrm{experiment}\mathrm{shows}\mathrm{that}4\mathrm{circles}$$$$\mathrm{can}\mathrm{produce}13\mathrm{closed}\mathrm{regions}(\mathrm{See}\mathrm{image}\mathrm{of}$$$$\mathrm{four}\mathrm{circles}\mathrm{above})\mathrm{while}\mathrm{your}\mathrm{formula}\mathrm{suggests}{2}^4-1=15$$$$\mathrm{Is}\mathrm{there}\mathrm{any}\mathrm{mistake}\mathrm{in}\mathrm{experiment}?$$

Commented by Yozzii last updated on 27/Dec/15

$$DownloadtheapplicationcalledGeoGebra.$$$$Iusedittoexperiment.Icouldcertainly$$$$bewrong,sousetheapptoseefor$$$$yourselfhowyoucanobtain15.I$$$${wasn}^{\prime }tsuccessfulinobtaining15$$$$onmyown.$$

Commented by Rasheed Soomro last updated on 27/Dec/15

$$\mathrm{The}\mathrm{process}\mathrm{should}\mathrm{be}\mathrm{as}\mathrm{follows}.$$$$\mathrm{The}\mathrm{second}\mathrm{circle}\mathrm{must}\mathrm{cut}\mathrm{the}\mathrm{first}$$$$\mathrm{circle},\mathrm{the}\mathrm{third}\mathrm{circle}\mathrm{must}\mathrm{cut}\mathrm{first}$$$$\mathrm{and}\mathrm{second}\mathrm{circle}\mathbf{at}\mathbf{distinct}\mathbf{points}.$$$$\mathrm{In}\mathrm{this}\mathrm{way}\mathbf{each}\mathbf{next}\mathbf{circle}\mathbf{must}\mathbf{cut}$$$$\mathbf{all}\mathbf{the}\mathbf{previous}\mathbf{circles}\mathbf{at}\mathbf{distinct}\mathbf{points},$$$$\mathrm{in}\mathrm{order}\mathrm{to}\mathrm{get}\mathrm{maximum}\mathrm{number}\mathrm{of}$$$$\mathrm{closed}\mathrm{regions}.\mathrm{I}\mathrm{think}\mathrm{you}\mathrm{have}\mathrm{done}\mathrm{well}.$$

Commented by prakash jain last updated on 27/Dec/15

$$\mathrm{I}\mathrm{will}\mathrm{try}\mathrm{experimenting}\mathrm{myself}.$$$$\mathrm{In}\mathrm{a}\mathrm{way}\mathrm{you}\mathrm{are}\mathrm{saying}\mathrm{a}\mathrm{venn}\mathrm{diagram}$$$$\mathrm{is}\mathrm{not}\mathrm{possible}\mathrm{with}4\mathrm{sets}\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\mathrm{to}\mathrm{show}$$$$\mathrm{all}\mathrm{of}\mathrm{the}\mathrm{following}.$$$$\mathrm{individual}4$$$$2\mathrm{pairs}6$$$$3\mathrm{paris}4$$$$\mathrm{all}\mathrm{intersection}1$$$$\mathrm{I}\mathrm{will}\mathrm{do}\mathrm{some}\mathrm{more}\mathrm{study}\mathrm{to}\mathrm{recall}.$$$$\mathrm{You}\mathrm{may}\mathrm{be}\mathrm{right}.\mathrm{I}\mathrm{may}\mathrm{have}\mathrm{forgotten}$$$$\mathrm{a}\mathrm{few}\mathrm{things}.$$

Commented by Yozzii last updated on 27/Dec/15

$$Thediagramoutliningallpossible$$$$eventsfor4setsdoesnotutilise$$$$acircle,asIsawinabookIlooked$$$$atrecently.Somethinglookinglike$$$$awrenchwasused,notacircle.{That}^{\prime }swhyI$$$$doubtedemployingf\left(n\right)=2^n-1.$$$$Ithinkthatifacirclewerepossible$$$$theauthor{wouldn}^{\prime }thaveusedthat$$$$wrenchshapetoyieldall16regions.$$

Commented by prakash jain last updated on 27/Dec/15

$$\mathrm{You}\mathrm{are}\mathrm{correct}n\mathrm{circle}\mathrm{can}\mathrm{at}\mathrm{most}$$$$\mathrm{creat}=n^2-n+2\mathrm{regions}.$$

Commented by Yozzii last updated on 27/Dec/15

$$Includingexternalregionright?$$$$Includingexternalregionright?$$

Commented by prakash jain last updated on 27/Dec/15

$$\mathrm{Yes}.$$$$\mathrm{Yes}.$$

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