Chords between six points
Six points are placed on a circle so that no three of the chords joining them pass through the same interior point. All 15 chords are drawn.
Into how many regions is the disc divided?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
With 1, 2, 3, 4, 5 points the counts are 1, 2, 4, 8, 16. Do not trust the pattern.
Draw the chords one at a time. A new chord adds one region, plus one more for every chord it crosses.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Add the chords one by one. A chord that crosses earlier chords is cut into pieces, and each piece splits a region in two. So the number of regions is 1, plus the number of chords, plus the number of crossings.
Step 1: chords. Six points give chords.
Step 2: crossings. Two chords cross inside the disc exactly when their four endpoints alternate around the circle. Any 4 of the 6 points give exactly one crossing pair, so there are crossings (no three chords meet at a point, so they are all distinct).
Step 3.
The trap. The sequence 1, 2, 4, 8, 16 for one to five points suggests 32. The formula only agrees with up to . With a regular hexagon the answer is 30, because three long diagonals meet at the centre.