KS3/4 Spacing Points Investigation Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 6th February 2013

Points in a unit square 1 Click to Reveal 1 You have to put 4 points into a square of unit side (i.e. length 1), such that the distance between the closest two points is maximised. What is the optimal arrangement, and what is this minimum distance? 1 Click to Reveal 1 The smallest distance between any two points is 1. We clearly can’t make this smallest distance any greater.

Points in a unit square √2 1 Click to Reveal 1 You have to put 2 points into a square of unit side (i.e. length 1), such that the distance between the closest two points is maximised. What is the optimal arrangement, and what is this minimum distance? 1 Click to Reveal √2 1 Clearly putting the points at opposite corners will maximise the distance.

Activity Now try 3 to 10 points. Remember you’re trying to maximise the minimum distance between any two points, i.e. the points are as spaced out as possible. For each: Show the optimal arrangement of points (solutions on next slides). Determine the smallest distance, giving the exact answer (e.g. in surd form) where appropriate). Number of points Distance 2 2 =1.414… 3 6 − 2 =1.035… 4 1 5 2 2 =0.707… 6 13 6 =0.601 7 2 2− 3 =0.536… 8 6 − 2 2 =0.518… 9 1 2 =0.5 10 0.421… ? ? ? ? ? ? ?

3 points ? 1 6 − 2 =1.035… 15° 1 Can use trigonometry on the triangle at the bottom. Can get value in surd form by using something called the ‘half-angle formula’ (which is covered in C3 at A Level).

4 points ? 1 Trivial.

5 points ? 2 2 1 Trivial.

6 points ? 13 6 =0.601 1 Trivial.

Consists of a square and two equilateral triangles. 7 points ? 2(2 – 3) 1 Consists of a square and two equilateral triangles.

Consists of a square and four equilateral triangles. 8 points ? 2(2 – 3) 1 Consists of a square and four equilateral triangles.

9 points ? Trivial.

Not that it’s not quite diagonally symmetrical! 10 points ? Not that it’s not quite diagonally symmetrical!

Extension Problems A large number of circular coins are arranged on a table in a square lattice, such that the coins are touching but not overlapping. What fraction of the table is occupied by coins? (You can ignore what happens at the edges of the table) What about if the coins are arranged in a hexagonal pattern? Can you give an argument why this pattern makes optimal use of the space?