Depth · one concept from first principles · step 2 of 4

Lines all crossing make the most points

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When $9$ line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, $1$ segment divides the circle into $2$ regions. In the 2nd figure, $2$ segments are drawn so they cross each other, dividing it into $4$ regions. In the 3rd figure, $3$ segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Operations

Your answer

How to solve

Strategy Look for a Pattern — Jumping straight to 9 segments is hard to picture, so I start with the small cases shown (1, 2, 3 segments), count how many new regions each fresh segment adds, and look for the rule. The pattern of 'extra regions added' turns out to be very simple, so I can extend it all the way to 9.

1STEP 1

Count the small cases by drawing

Drawing them out: 1 segment gives 2 regions, 2 give 4, and 3 give 7.

1 seg→ 2, 2 seg→ 4, 3 seg→ 7

2STEP 2

See how many regions each new segment adds

The k-th segment crosses the k-1 before it, so it splits into k pieces and adds k regions.

2→4 adds 2, 4→7 adds 3

3STEP 3

State the rule

So the total is 1 for the whole circle plus 1 + 2 + 3 and on up to the number of segments.

R(n)=1+(1+2+3+…+n)

4STEP 4

Add up to 9 segments

The circle counts 1, and 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45, so the total is 46.

R(9)=1+(1+2+…+9)=1+45=46

Answer

46 regions

The counts climb 2, 4, 7, 11, 16, ... — always going up by one more than the time before, which matches segments that meet ever more crossings. The answer 46 is a whole number of regions, much bigger than the 7 we get for 3 segments, which makes sense for 9 segments.

Takeaway

Each new line that crosses all the others adds as many regions as its own number, so just add 1+2+3+...+9 and one for the circle.

Count the small cases by drawing
See how many regions each new segment adds
State the rule
Add up to 9 segments

Where next?

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