Monday, April 6, 2015

[H2PCRP20150406] Colouring The Pentagon (Combinatorics)



Introduction
     This is a question suitable for the mainstream Junior College students taking H2 mathematics, but is some primary school olympiad question from somewhere.  Whatever!  Mathematics is for everybody, young and old.  Anybody can solve this problem if  s/he makes observations and uses the right approach and thinking skills.

An Incisive Insight
     Although this pentagon is not a regular pentagon, the colouring scheme depends just on the order of colours on the edges.  We can start from one edge and see what colours are possible.  And then we can rotate the colouring scheme around.  [ We are breaking down and simplifying the problem. ] 
     Fiddling around with various possibilities, you might realise that:-
·  you cannot have three of the same colour going round the pentagon
·  you cannot have three sets of pairs of edges with the same colour.
·  you cannot two single colours and one double colour
There must be one single colour and two pairs of doubled colours.  All colouring schemes will have a  “12123”  colouring pattern going around in a loop.  The diagram below shows an example where colour 1 = yellow (Y), colour 2 = red (R)  and colour 3 = blue (B).


Do we need to consider a “21213”  pattern?  If  “12123”= “YRYRB”, we can later reassign colours, swapping R and Y to give 1=R and 2=Y and then “12123”=“RYRYB”.  So we have got that covered.  Let us worry about the reassignment later. 

Solution
     Observe that the position of the “3” (the single colour) can be rotated round the edges of the pentagon in  5  ways.

     Observe also that there are  3! = 3 ´ 2 ´ 1 = 6  ways to shuffle the colours i.e. assign colours 1, 2 and 3  to  Y, R and B.

The above two processes (rotation and shuffling) are independent of each other.  Rotation of the single colour can be done with or without the shuffling of colours.  Hence we can use the Multiplication Principle and calculate
          the total number of ways = 5 ´ 6 = 30.
Tada!

H02. Use a diagram / model
H04. Look for pattern(s)
H09. Restate the problem in another way
H10. Simplify the problem
H11. Solve part of the problem

Suitable Levels
* GCE ‘A’ Level H2 Mathematics
* IB Mathematics HL / SL
* Primary School Maths Olympiad
* other syllabuses that include combinatorics

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