Emma’s Dilemma

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A.M.D.G                                                                                             June 2001

Emma’s Dilemma

I firstly found the different number of arrangements for Emma’s name.

  1. EMMA

  1. EMAM

  1. EAMM

  1. AEMM

  1. AMEM

  1. AMME

  1. MAME

  1. MAEM

  1. MEAM

  1. MEMA

  1. MMAE

I then found the different number of arrangements for Lucy’s name.

  1. LUCY

  1. LUYC

  1. LYUC

  1. LYCU

  1. LCUY

  1. LCYU

  1. ULCY

  1. ULYC

  1. UYLC

  1. UYCL

  1. UCYL

  1. UCLY

  1. CULY

  1. CUYL

  1. CYUL

  1. CYLU

  1. CLYU

  1. CLUY

  1. YLUC

  1. YLCU

  1. YCLU

  1. YCUL

  1. YUCL

  1. YULC

The method I used to investigate the number of different arrangements of Emma’s name was by keeping the first two letters the same and rearranging the last two letters. I kept on doing this until I had the maximum number of arrangements, which was 12. I repeated this procedure with Lucy’s name and found 24 arrangements.

Next I found the arrangements of some different names to see if I could spot a pattern.

  1. TOM

  1. TMO

  1. MTO

  1. MOT

  1. OMT

  1. OTM

My strategy for finding the number of arrangements for the name Tom, was by this time, keeping only the first letter the same and rearranging the last two letters. I kept on doing this until I had found the maximum number of arrangements, which was 6.

In order to get an orderly pattern, I then decided to find the arrangements of a name with two different letters and one letter.

  1. J

One letter gives just the one arrangement.

  1. JO

  1. OJ

Two different letters give two arrangements. I don’t think I need any more arrangements of names because I think you can get a simple formula from four names. I then arranged the four names in size order. I did not include Emma’s name because it has two of the same letters, which makes it irregular.

J-1

JO-2

TOM-6

LUCY-24

In order to find a pattern for the number of arrangements, I decided to start from the bottom.

Join now!

One pattern I found from this was that if you divide the number of arrangements by the number of letters of a name, you would get the number of arrangements of the previous name.

For example: Lucy, a four-lettered name, has 24 arrangements. 24/4=6, which is the number of arrangements for the name Tom. This works for all names except J because no letters gives no arrangements. Tom has 3 letters and 6 arrangements. 6/3=2, which is the number of arrangements of the name Jo.

This gives us a formula of:

p=a/l

p= number of arrangements ...

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