Investigation 9:- The Bullies.
Pete is a bully. Pete likes to shove his weight around. Pete
does not like to have to wait in queues, especially for his
lunch. Each day in the dining room when there is a queue of
pupils waiting to be served, Pete can be seen trying to push
his way to the front of the queue. Pete starts at the back
of the queue but every time he sees a pupil being served he
jumps past two pupils in the queue.
The diagram shows a queue of 12 pupils ( each pupil is marked
with an x ) and Pete ( marked with a ) is at the back of the
queue.
s
e
r
v
e
r
y
Queue here
Pete
x x x x x x x x x x x x
Show that 4 pupils will be served before Pete.
Investigate for different queue sizes and see how many pupils
are served before Pete in each case.
Bullies - Page 1 of 3
Record your results in a table like the one shown.
Number of pupils
in the queue (n)
Number served
before Pete (N)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
You will soon be able to spot a pattern in the table. You may
wish to extend the table beyond the 18 shown.
Can you use the results in the table to answer the following
question?
How many pupils (N) will be served before Pete if there are
initially
a) n = 80 pupils b) n =145 pupils
in the queue before Pete arrives?
The function Int[n] is defined as ‘Int(n) = smallest integer
larger than n’
Eg. Int[3.4] = 4 and Int[5.9] = 6.
Can you write the number N of pupils served before Pete in a
queue of length n, in terms of the Int function.
b) Bertha is an even bigger bully than Pete. She
does not hang around in queues. She jumps 3
places up the queue every time she sees a pupil
ahead of her being served.
Make up a table as before to record the number N of pupils
served before Bertha.
Bullies - Page 2 of 3
Use your table to calculate how many pupils (N) will be served
before Bertha if there are initially
a) n = 94 pupils b) n =150 pupils
in the queue before Bertha arrives?
Can you write the number N of pupils served before Bertha in
a queue of length n, in terms of the Int function.
c) Mabel is the ultimate bully who jumps 5 places every time
she sees a pupil being served. How many pupils are served
before Mabel if there are initially
(i) 57 pupils
(ii) 135 pupils
(iii) n pupils
in the queue before Mabel arrives.
Bullies - Page 3 of 3