Solution 2 (Eigenvalue): Monkeys & Coconuts

Math Online Tom Circle

Solution 2: Use Linear Algebra Eigenvalue equation: A.X = λ.X

A =S(x)= $Latex \frac{4}{5}(x-1)$  where x = coconuts

S(x)=λx

Since each iteration of the transformation caused the coconut status ‘unchanged’, which means λ = 1 (see remark below)

$Latex \frac{4}{5}(x-1)=x$
We get
x = – 4

Also by recursive, after the fifth monkey: $Latex S^5 (x)$ = $Latex (\frac{4}{5})^5 (x-1)- (\frac{4}{5})^4-(\frac{4}{5})^3- (\frac{4}{5})^2- \frac{4}{5}$

$Latex S^5 (x)$ = $Latex (\frac{4}{5})^5 (x) – (\frac{4}{5})^5 – (\frac{4}{5})^4 – (\frac{4}{5})^3+(\frac{4}{5})^2 – \frac{4}{5}$

 

$Latex 5^5$ divides (x)

Minimum positive x= – 4 mod ($Latex 5^{5}$ )= $Latex 5^{5} – 4$= 3,121 [QED]

 

Note: The meaning of eigenvalue  λ in linear transformation is the change  by a scalar of λ factor (lengthening or shortening by λ) after the transformation. Here

λ = 1 because “before” and “after” (transformation A)  is the SAME status (“divide coconuts by 5 and left 1”).

View original post

About mathtuition88

http://mathtuition88.com
This entry was posted in maths tuition, Uncategorized and tagged , . Bookmark the permalink.

1 Response to Solution 2 (Eigenvalue): Monkeys & Coconuts

  1. tomcircle says:

    See the best solution (3) using just Singapore PSLE Modelling Math:
    http://tomcircle.wordpress.com/2013/03/30/solution-3-best-monkeys-coconuts/

    Like

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.