Maths IA · Algebra and functions · topic 5 of 40
Chords in a circle: which formula counts the regions? · IB Maths IA idea: guide, sample graph and step-by-step plan
a classic: what makes it yours?Join points on a circle and count the regions. 1, 2, 4, 8, 16... you think you know what comes next. Draw one more point and check.
- ✓ you like patterns that turn out to be tricky
- ✓ you can draw carefully and count systematically
- ✓ you like proving things, not only guessing them
The big idea
← the broken pattern is your hookJoin n points on a circle by chords, with no three chords meeting at a point. The first few counts look like a doubling pattern, then the pattern breaks. Your job is to find a closed form that really fits and then prove it: by finite differences and polynomial fitting, by counting crossings, or by induction. The best explorations explain why the doubling pattern fails, not only that it does. At HL, fitting a polynomial and stating it is not enough for criterion E: you need a proof, and ideally a link to binomial coefficients or Euler's formula.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Show that each new chord adds regions equal to one more than the number of crossings it makes, then add up.
→ You get the formula from the geometry itself, using combinations, and your proof is not an induction on a guess.Take your fitted formula and prove it by induction, adding one point at a time.
→ The core is a rigorous inductive step, and you must justify the general-position assumption you used.Treat the picture as a planar graph (vertices, edges, faces) and count V, E and F. Euler's formula for planar graphs is not in AA, so introduce and justify it.
→ A different proof that links geometry and graph theory, with a check against your first method.Allow three chords to meet at a point, for example with points evenly spaced.
→ The count drops. You must say by how much and why, which turns the assumption into a result.Count only the crossing points, or only the triangles in the picture, instead of the regions.
→ A smaller counting problem that you can solve completely and compare with the regions count.Cutting a pizza or a cake with straight cuts, or lines in a plane instead of chords in a circle.
→ A real-world introduction, with a related sequence to compare. The maths changes slightly, so say how.Your Maths IA research question: run the check
what exactly are you proving?Write your own question first. Then make sure it ticks every box:
- Says exactly what is counted (regions of the disc, with points in general position) and what the variable is (number of points, or number of lines).
- States what you will produce: a closed form, a proof of it, or both.
- Can be answered by reasoning you show, not only by fitting a polynomial with software.
- Says what you will compare your result with, such as the doubling pattern or a second proof.
- Is narrow enough to prove properly within 12 to 20 pages.
The picture to draw
Six points, every pair joined. Draw your own, and mark the crossing points: the whole proof can be built from counting them.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
The doubling pattern you expect, why you doubt it, and your aim in your own words.
Clear diagrams for small n, a systematic way to count regions, and the general-position assumption stated and drawn.
A table of counts, the failure of the doubling guess, and finite differences or another method to find a candidate form.
The method you use to get a formula (differences or polynomial fitting) with the working shown, and a check with a count you did not use.
Induction, crossing counts or Euler's formula. The key steps justified, with the role of general position explicit.
Another proof, or what happens when three chords meet. Compare the results.
What the proof needs, where the formula stops applying, and what you changed after your first guess failed.
Answer your aim and say what you now believe about the doubling pattern.
Sources for the problem and the proof technique.
What your Maths IA graph could look like
a curve is not a proof!Made-up data, to show you the form of a strong graph, not the answer. Your counts come from your own drawings. example only
- Axis labels name what is counted and what is varied. Whole numbers only, so say that the curve is a model, not extra data.
- Points are your counts, drawn exactly. No error bars: the only error is a miscount, and your table should be checked.
- The doubling curve is drawn too, so the reader sees how the wrong guess fails.
- The scale runs far enough to show the break clearly, while still showing every point.
- Another graph to consider: plot the differences between successive counts, and the differences of those, to show how the degree of the polynomial appears.
Your game plan
draw n = 5, 6, 7 firstTick steps off as you go. We'll remember on this device.
0 of 10 done
- Draw and countDraw small cases by hand with points spread irregularly. Count regions carefully. Decide in advance how you will count so that nothing is missed or counted twice.
- PredictWrite the next count that the doubling guess gives. Then draw it. What do you find, and what does that make you think?
- Trial run (1 lesson)Draw n = 5, 6 and 7 with great care. Check each count with a second person or a second method. Fix any mistakes before you build on them.
- Choose a way to find a formulaFinite differences, polynomial fitting, or counting the crossings. Write down why you chose it and what it assumes.
- Find a candidateWork it out with the steps shown. Test it on a case you did not use to build it.
- Decide what you will proveSay exactly what the claim is, for all n, and which method suits it best. Check the role of the general-position assumption.
- Prove itWrite the proof in clear steps, each justified. If you use induction, say what the inductive step adds and why.
- Second viewDo a second proof, or relax an assumption and see what happens. Compare, and say which you trust more.
- ReflectWhere did your first idea fail? What part of the proof surprised you? What would you test next?
- Write, cut, checkWrite in the layout order, with one picture per idea. Check the checklist, then get your question checked before your teacher's draft.
Maths you'll need
Traps to avoid in this Maths IA
- Fitting a polynomial with software and calling it the answer. A fit through seven points is a guess until it is proved.
- Drawing the points evenly spaced, so three chords meet and the count is lower than the general case.
- Miscounting regions in a crowded picture. Count twice, with a method.
- At HL, stopping at pattern and polynomial fitting. Without a proof or a link between areas of maths, criterion E stays at the middle band.
- Using the formula to predict a case and never testing it on one you drew.
- Pages of pictures for every n. Show two or three and put the rest in a table.
- A history of Moser's problem and nothing of your own thinking.
More Maths IA ideas like this
- Newton's method: how fast, and when does it fail?●●○
- Repeating decimals: what is the pattern in the digits?●●●
- Roots of unity: a pentagon by ruler and compass?●●●
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.