Waiters’ Nightmare
 A single rectangular table can seat 6 hungry humans
Waiters’ Nightmare
 However, two rectangular tables can seat various numbers of
hungry humans depending on the layout
 You need to find the
minimum and
maximum number of
hungry humans that
can be fed, using
different numbers of
tables
Number of
Tables
2
3
4
5
6
10
Smallest
Number of
Seats
Largest
Number of
Seats
Your Mission
1. Complete the layouts of seats on the diagram for two tables
2. Fill in the results table
3. Draw as many different layouts as possible for three tables. Add the
seats, fill in the table with your results.
4. Repeat for layouts of four tables and five tables.
Investigation Tips
Extension Tasks




1. Try and identify the term to term
rule for the smallest number of
seats
2. Repeat for the largest number of
seats
3. Use these rules to predict the
numbers for 6 & 7 tables.
4. Draw out the tables for 6 & 7 to see
if your predictions were right.
Clearly presented.
Draw diagrams to help you.
Use a pencil and ruler.
Record your results in the
table as you go.
 Look for patterns.
The Staircase Investigation
 This is the start of a sequence…
 How many cubes do you need for the 5th staircase?
 How many cubes for the 6th staircase?
 Can you find a rule linking the pattern number to the number of
cubes used?
 Use your rule to find how many cubes you would need for the 10th
staircase.
The Staircase Investigation
 This is the start of another sequence…
 How many cubes do you need for the 4th and 5th staircase?
 Can you find a rule linking the pattern number to the number of
cubes used? Test your rule…
 Use your rule to find out how many cubes you need for the 10th
staircase.
The Staircase Investigation
 Here is yet another type of stairway sequence…
 How many cubes do you need for the 4th and 5th staircase?
 Can you find a rule linking the pattern number to the number of
cubes used? Test your rule…
 Use your rule to find out how many cubes you need for the 10th
staircase.
Hints….
Hints…
These are some quick ways you can use to count the total cubes used each
time
Hints…
Handshake Investigation
If everybody in this classroom today shook hands with every other
person, how many handshakes would there be?
Begin by making a table and testing it out for 1 person, 2 people,
3 people, etc.
You may like to try it out amongst you.
Try and spot any patterns you may notice.
There are a few different methods you can use to organise your
working and record your results. Ask me for hint sheets if unsure.
Hints…
Bunny Rabbits Investigation
• In 1202, Fibonacci investigated about how fast rabbits could
breed in ideal circumstances.
• One male and one female rabbit are put in a field after birth.
• Rabbits can mate at the age of one month, and it takes one
month until the new bunnies are born.
• Hence, at the end of its second month a female can produce
another pair of rabbits.
• Assume your rabbits never die and that the female always
produces one new pair (one male, one female) every month
from the second month on.
• How many pairs will there be in one year?
Hints…
Bunny Rabbits Investigation
At the end of the first month, they mate, but there is still one only 1 pair.
At the end of the second month the female produces a new pair, so now there
are 2 pairs of rabbits in the field.
At the end of the third month, the original female produces a second pair,
making 3 pairs in all in the field.
At the end of the fourth month, the original female has produced yet another
new pair, the female born two months ago produces her first pair also,
making 5 pairs.
The number of pairs of rabbits in the field at the start of each month is
1, 1, 2, 3, 5….
Hints…
Hailstone Sequence Investigation
Hints…
Hailstone Sequence Investigation
Investigation 1
Collatz believed that whatever number you start with, you will always end up
with a 1. However, this has not been proven yet.
Try out different ‘seeds’ and see if you agree or disagree with Collatz. Why?
How long does it take for each of your sequences to reach 1? Can you predict
the length of the sequence from the ‘seed’ chosen?
Create at least one graph, with three different seeds,
to show how the numbers ‘bounce’ around
like a hailstone.
Your graphs should look similar to this one 
Hints…
Hailstone Sequence Investigation
Investigation 2
Hints…
Hailstone Sequence Investigation Hints
Hints for Investigation 2