Scribd
Upload
56 views1 page

Problem of The Month: Webpage: Cemc - Uwaterloo.ca/resources/potm - PHP Email: Potm@uwaterloo - Ca

1) There are k dials initially showing 1 at the top that can be rotated in pairs up to 4 positions. The number of possible configurations is 4^k. 2) There are k dials labelled 1 to n that can be rotated in pairs. The number of possible configurations is n^k. 3) With the additional move of rotating the leftmost and rightmost dials, the number of possible configurations is even greater than just n^k.

Uploaded by

ElevenPlus Parents
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
56 views1 page

Problem of The Month: Webpage: Cemc - Uwaterloo.ca/resources/potm - PHP Email: Potm@uwaterloo - Ca

1) There are k dials initially showing 1 at the top that can be rotated in pairs up to 4 positions. The number of possible configurations is 4^k. 2) There are k dials labelled 1 to n that can be rotated in pairs. The number of possible configurations is n^k. 3) With the additional move of rotating the leftmost and rightmost dials, the number of possible configurations is even greater than just n^k.

Uploaded by

ElevenPlus Parents
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 1

Problem of the Month

Problem 2: November 2019

(a) Several dials are each labelled with the integers from 1 through 4 in
clockwise order. The dials are arranged in a row and initially configured so
that each dial shows 1 at the top:

1 1 1 1 1
···
4

2
3 3 3 3 3

A move consists of rotating two adjacent dials in the same direction by the
same number of positions. For example, one possible move is to rotate the
second and third dials by one position in the counterclockwise direction
resulting in the configuration shown:

1 2 2 1 1
···
4

2
3 4 4 3 3

There are k ≥ 2 dials and they are initially configured so that each shows 1
at the top. In terms of k, determine how many possible configurations of the
dials are attainable by performing a sequence of moves.
(b) Suppose now that there is an integer n ≥ 2 so that each of the k ≥ 2 dials is
labelled in clockwise order by the integers 1 through n beginning with 1 at
the top. Find the number of configurations of the dials that are attainable
by a sequence of moves. Your answer should be in terms of n and k. Again,
a move consists of rotating two adjacent dials in the same direction by the
same number of positions.
(c) Answer the question in part (b) with the following additional type of move
allowed: rotate the leftmost and rightmost dials in the same direction by the
same number of positions.

Webpage: cemc.uwaterloo.ca/resources/potm.php Email: potm@uwaterloo.ca

You might also like