Senior Challenge 4
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here to open a pdf version for printing if you need it. There are applets
for Questions 2 and 3. These can be activated by clicking on the appropriate
diagrams in the questions.
Question 1: Naturally Mathematical
Sequences
| In a Fibonacci sequence, each term
is made by adding the previous two terms. The original Fibonacci
sequence starts with 0 and 1 and the first six terms are:
0, 1, 1, 2, 3, 5,
Naturally mathematical
sequences are made in the same way, but can start with any two
numbers (so long as these are not negative). For example, a
naturally mathematical sequence could start with 3 and 7, and then
the first six terms would be:
3, 7, 10, 17, 27, 44,
What starting numbers in a naturally
mathematical sequence lead to the sixth term being 66? |
Fibonacci
(Leonardo of Pisa) |
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Question 2: Patterns on a Grid
In each of the patterns A F there is one
coloured square in each row and one in each column of the 3ื3 grid.
The Starter
Two patterns are only called different if
you cannot rotate and/or flip one to make it match the other. How many
different patterns are there in the A F patterns and which ones are the
same?
The Challenge
How many different patterns can you make
on a 4ื4 grid?
(Hard) How many different patterns can you
make on a 5ื5 grid?
Click on the above diagram to open an
applet that will help you explore the possibilities for both parts of this
question. |
Question 3: Common Totals
The numbers 1 - 10 have been positioned to make a
rectangle and totals of the rows and columns have been calculated.
Thats the easy part, because we have
provided an applet to help you both position the numbers and calculate
those totals. Click on the diagram above to launch the applet in which you
can swap two numbers by clicking on them.
By moving the numbers to different
positions, it is possible to make all the totals the same.
The Challenge
What are the common totals that are
possible?
Give one example of how each one is made.
Suggestion
Rather than just guess and check to
find possible common totals, you could add the numbers 1 10 and then add
in the numbers at the corners because they are used twice, once in a row
and once in a column, to make the common totals.
That will tell you what the row and column totals add to, and, if they are
all to be the same, then there is something special about that combined
total. You will now be able to know in advance what the corner numbers
will add to if you are trying to make the common totals equal. |
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