Math Is Fun Forum

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

dots and regions

I've drawn six circles (any size will do).
The first has a single dot on the circumference and the inside has one region.
The second has two dots joined togther by a straight line. This divides the inside of the circle into two regions.
The third has three dots on the circumference and every dot is joined to every dot. This makes four regions.
The fourth has four dots with every dot joined once to all the other dots. This divides the circle into eight regions.
The fifth has five dots. It was getting difficult to fit in the numbers to count the regions, so I'll leave it as an exercise for you. You can make a bigger circle if it helps. Fill in the missing number of regions.

Now make a prediction. How many regions will six dots make?

Once you have an answer, draw it to see if you were correct.  It is possible to put the dots so that three lines are concurrent.  If this happens you will lose a region, so I suggest you reposition one dot so the lines that cross separate and this will open up another region.

Was your prediction correct?

Please post back, hiding your answer with hide tags.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,884

Re: dots and regions

Hi Bob;

Nice problem!

Last edited by phrontister (2024-03-10 18:05:42)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: dots and regions

hi Phro,

Excellent!  It's a problem I do with most of my maths classes at some time, a part of my topic of "The Nature of Proof".  You can do it at any age and with very little prep.

I think it's important as it shows that just getting a recognisable sequence is not enough to be certain of the 'formula'.

Other readers: Don't look up the answer; try it for yourself; there's useful learning lurking here.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob