Counting in Geometry

Transcript

0:00Introduction

Hello everyone! Okay, one small question before we start. How many squares are there on a chessboard? Sixty four, right? Eight rows, eight columns. Hmm, look again. The whole board is also a square. Every block of four little squares is a square too. So the true answer is much bigger than sixty four! And the rectangles? Even more. Today we count them all, without drawing any. The secret is one idea: in geometry, counting is just choosing. Questions like these come up often in JEE, so let's begin.

0:36Points: choose two for a line, three for a triangle

So, let's start with points. Take n points in a plane, with no three of them collinear, that is, no three on one straight line. Now tell me, how many points do you need to fix a straight line? Two, yes, exactly! Any two points fix a line, and their order does not matter, so this is a selection. The number of lines is n C two: the capital C, with n at its top left and two at its bottom right. The same way, any three points give a triangle, so n C three triangles. Right? Now, a twist. Suppose exactly m of the n points lie on one straight line. How many different lines do those m points give? Just one! But n C two counted m C two pairs from them, all giving the same line. So subtract m C two, and then add back that one line: n C two, minus m C two, plus one. Forgetting this plus one is the classic trap. Don't do that, okay? Now triangles. Three collinear points give no triangle, just a flat line. So subtract m C three: n C three, minus m C three. And no plus one here, because a straight line is not a triangle. Clear?

1:51Polygons: diagonals, crossings, triangles

Now, polygons. On screen is a hexagon with six vertices, labelled A one to A six. The small number at the bottom right is just a label. Join any two vertices, and you get either a side or a diagonal. Six C two is fifteen joins in all. Six of them are sides, so the other nine are diagonals, the lines inside it. That is the top line next to the figure. Okay? Now, take a convex n gon, a polygon with n sides and no corner pointing inwards. Same idea. Can you tell me how many diagonals it has? Very good! All the joins, minus the sides: n C two, minus n. And that simplifies to n times, n minus three, over two. With me? Now, where do the diagonals cross inside the polygon? Pick any four vertices. How many crossing points do they give? Exactly one! Four vertices make a quadrilateral, and its two diagonals cross once, inside. So there are n C four crossings inside; for the hexagon, six C four, fifteen, as the figure says. But read the small note under it: in the regular hexagon, the three long diagonals all meet at the centre. So the formula needs a condition: no three diagonals concurrent, that is, no three through one point. Got it? Now triangles from the vertices. The figure shows six C three, twenty in all. But look at the red triangle, with vertices A one, A three and A five. It uses no side of the hexagon. The number of triangles with no side of the polygon is n times, n minus four, times, n minus five, over six. Okay? And triangles with exactly one side of the polygon. Choose that side: n ways. The third vertex cannot be either end of that side, or the vertex just beyond either end, since that would make a second side. So n minus four choices. That gives n times, n minus four.

3:45Grids, parallel lines and lines in a plane

Right? Alright, now grids. Draw a rectangle on squared paper, along the lines. What do you need to choose to fix it completely? Two horizontal lines, and two vertical lines. Yes, exactly! Choose them, and the top, bottom, left and right sides are all fixed. Clear? Now take a grid of unit squares, with m rows and n columns. How many horizontal lines does it have? Just m? No, careful! m plus one. There is always one more line than rows of squares, so count lines, not squares. On screen, m cross n means m rows and n columns, and the small dot means times. So the grid has m plus one C two, times n plus one C two rectangles. Okay? Same idea for parallelograms. p parallel lines crossing q other parallel lines: choose two lines from each set. So p C two, times q C two parallelograms. With me? Last one. Now n lines in a plane, with no two parallel, and no three concurrent. Parallel lines never meet, and three lines through one point make no triangle. That is why we need both conditions. Right? Then any two of the lines meet in exactly one point, so n C two intersection points. And any three of them make a triangle, so n C three triangles. Again, just choosing. Clear?

5:12Example 1: lines and triangles from 12 points, 5 collinear

Now, example one. Twelve points lie in a plane, and exactly five of them are collinear. How many lines, and how many triangles, do they determine? Pause the video, and try it first. Done? Okay, let's check. If no three were collinear, twelve C two: sixty six lines. Right? But the five collinear points give five C two, that is ten pairs, all on the same line. So the answer is sixty six minus ten, fifty six lines, isn't it? No, no, wait! That is the trap. The line through those five points is still there, and we just removed it completely. So add it back once. Sixty six minus ten, plus one: fifty seven lines. Clear? Now triangles. Twelve C three is two hundred and twenty. Remove the triples taken from the five collinear points: five C three, ten. So two hundred and ten triangles, and no plus one here. Got it?

6:11Example 2: rectangles and squares on a chessboard

Now, remember our chessboard from the start? Let's finish it. How many rectangles, and how many squares, are there on an ordinary chessboard? On screen, eight cross eight means eight rows and eight columns. Pause and try. Hint: count lines, not squares. Done? Okay, let's check. Rectangles first. We choose two horizontal lines and two vertical lines. So, how many horizontal lines does the board have? Nine, correct! Eight rows of squares, so nine lines. And nine vertical lines too. If you said eight, that is the trap we just saw. Okay? So the rectangles: nine C two, that is thirty six, in brackets, and squared: the small two at the top right. Thirty six squared is one thousand two hundred and ninety six. That many rectangles! Right? Now the squares. Take a square of side k, with k from one to eight. It fits in nine minus k places across, and nine minus k places down. So nine minus k, the whole squared. Clear? Side one gives eight squared, sixty four, the ones everybody counts. Side eight gives one squared, just the whole board. Add them all: one squared, plus two squared, and so on up to eight squared. The three dots mean and so on. Two hundred and four squares! See, mystery solved: not sixty four, but two hundred and four. Got it?

7:36Quick revision: lines, triangles, diagonals, rectangles

So, quick revision. For a line, choose two points; for a triangle, choose three. With collinear points, subtract; and for lines only, add the one line back, never for triangles. In a polygon: n C two minus n diagonals, and n C four crossings inside. For rectangles, choose two horizontal and two vertical lines, and count lines, not squares. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!