1. Problem of Investigation

2. 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.

3. Problem of Investigation The diagram shows a queue of 12 pupils: serveryservery x x x x x x Queue here Pete is here

4. Problem of Investigation Investigate for different queue sizes and see how many pupils are served before Pete in each case. Record your results in a table like the one shown. Number of pupils in the queue (n) 23456789101112131415161718 Number served before Pete (N) 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?

5. Problem of Investigation 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.

6. Problem of Investigation 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.

7. Problem of Investigation 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.

8. Problem of Investigation 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.