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'Music to My Ears' printed from https://nrich.maths.org/
Rhiannon from St Mary Redcliffe Primary
School sent a very well-explained solution to this problem. She
wrote:
On the 15th beat, you will be doing a click. I know this because 15
is in the three times table and there is a click every three
beats.
On the 20th beat, you will be doing a
clap, because 20 is not in the three times table.
On the 99th beat you will be doing a
click, because 99 is in the three times table. I know this because
9+9 = 18 and 1+8=9 and 9 is in the three times table.
On the 100th beat you will be doing a
clap because you did a click on the 99th beat.
If two people do the following
rhythms:
person 1: clap clap click
person 2: click clap clap
then you won't click at the same
time.
On person 1, you click if the number is
dividable by three
(we can say "divisible
by 3") .
On person 2, you click if the number has
remainder 1 when divided by three. Since a number can never ever be
dividable by three and remainder 1, you won't click at the same
time.
To have the same result, person 1 could
go "clap click clap", person 2 could go "clap clap click".
So that you do click at the same time you
could:
person 1: clap clap click
person 2: clap clap clap click
You would both click at the same time every 12 beats.
Fantastic, Rhiannon!
Rohan from Longbay Primary School suggested an alternative:
Change your friend's rhythm to: click
clap click etc. and keep your rhythm the same. 9.
A click on your rhythm occurs every 3rd
beat and a click on your friend's rhythm will occur every 1st and
3rd beat so you will both click at the same time on the 3rd beat.
Well done to everyone who
sent in solutions.