MONKEY PUZZLE
3 monkeys spend all day collecting bananas. At the end of the day
they are very tired and go to sleep.
During the night, one of the monkeys wakes up and, feeling
hungry, eats a banana. He then decides to take one third of the
pile and hides it. He goes back to sleep.
Another monkey wakes up. He too is hungry, so eats a banana.
He also takes a third of the pile and hides it. He goes back to
sleep.
The last monkey wakes up, eats a banana and takes a third of the
pile.
In the morning when they are all awake they share the remaining
pile of bananas three ways.
Monkey puzzle
Saturday 21 November 1998
B Sharratt
Your tasks
 Discuss what arithmetic is necessary to work out the problem.
 Use a spreadsheet program, such as EXCEL, to show your
arithmetic. Set it out as follows:
Enter this number
=B5-1
=B8/3*2
=B9-1
=B12/3*2
=B13-1
=B16/3*2
=B17/3
 The number of bananas in the pile has to allow for whole
numbers of bananas to be taken by the monkeys. By changing
the number in the pile, find out the next highest number of
bananas in the pile.
 Repeat this to find the next 2 numbers in the sequence. Can
you see a pattern? What is your pattern?
 Use your pattern to make a list of the first six numbers. Check
each number by typing it into B5 on your spreadsheet (Number
to start).
 How can we find the 100th number in the pattern? Do we have
to find them all until we get to the 100th?
No!!!
Monkey puzzle
Saturday 21 November 1998
B Sharratt
How do we find the 100th term?
Here are the first six numbers in a list with their order at the side:
order
1st
25
+81
nd
2
106
+81
rd
3
187
+81
th
4
268
+81
th
5
349
+81
th
6
430
The pattern you have found is:
Number of bananas in the pile = 25 + 81 +81 + 81 …….
 How many 81s are added to 25 to get the second number
(106)?
Therefore:
Number of bananas in 2nd pile = 25 + (81 x 1)
 How many 81s are added to 25 to get the third number (187)?
Therefore:
Number of bananas in 3rd pile = 25 + (81 x 2)
 How many 81s are added to 25 to get the fourth number (268)?
Therefore:
Number of bananas in 4th pile = 25 + (81 x 3)
Complete the following:
Number of bananas in 100th pile = 25 + (81 x ?)
Monkey puzzle
Saturday 21 November 1998
B Sharratt
Number of bananas in 1001st pile = 25 + (81 x ?)
 How have we arrived at the number to multiply by?
If we want to write an equation that will allow us to find how many
bananas are in the pile we can use letters to represent numbers.
We will use ‘n’ to represent the 1st, 2nd, 3rd, 4th etc.
You will have noticed that the number we multiply ’81’ by is one
less than the position in the list. Eg the 100th term is 25+(81*99)
Therefore in terms of ‘n’ we can replace the equation with ‘n-1’.
The equation looks like this:
nth term = 25 + 81*(n-1)
or
nth term = 25 + 81(n-1)
This can be simplified to:
25 + 81n - 81
=81n – 56
Therefore
nth term = 81n – 56
 Use this equation to find the number of bananas in the 201st
pile.
 Test it in your spreadsheet.
 Use the equation to find the number of bananas in the 1001st
pile
 Test it in your spreadsheet.
 Does this work with other numbers of your choice?
Monkey puzzle
Saturday 21 November 1998
B Sharratt