Download Playing with Patterns

Playing with
Patterns
Andrew Derer
MathScience Innovation Center
Richmond, Virginia
Using Student
Responders
To respond to a
question:
• Wait for polling
to be open.
• Select your
response while
aiming at the
receiver.
Aim this…
…at this
From the list below, which
is your favorite class?
79%
11% 11%
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M
1. Math Class
2. Science Class
3. Social Studies
Class
4. Language Arts
Class
5. Any class with
math in it
What is encryption?
47%
26%
16%
11%
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1. Mixing up numbers or
letters in no specific
order.
2. Making a pattern which
is impossible to decode
without the key.
3. A special system used
by spies to get into other
countries.
4. A new computer running
on the Intel Pentium VI
chip.
5. All of the above.
What is a cryptex?
• A cryptex is a device credited to Leonardo
DaVinci which can hold a scroll or secret
note.
• It cannot be opened without the correct
code.
• The one in the picture below has 5 dials
and 26 letters per dial.
• This means there are approximately
11,800,000 combinations.
Encryption is…
• A defined pattern which allows users to
make information useless unless decoded
properly.
• A method of securing information for
transport either physically or electronically.
• A defined topic in mathematics which has
helped make internet commerce possible.
Finding the Code
• Your task today is to unlock the code and
retrieve the prize.
• You will work in groups, but open the
cryptex as a class.
• Each pattern will reveal another piece of
the puzzle.
• Good Luck!
• This slide will self destruct in
10 seconds.
What is the next piece in
the following sequence?
2, 5, 9, 14, _____
2, 5, 9, 14, __?__
0%
0%
100%
0%
0%
0%
0%
1.
2.
3.
4.
5.
6.
7.
18
19
20
21
22
28
30
What is the next piece in
the following sequence?
20
2, 5, 9, 14, _____
How about this one?
,
,
,
,
Answer?
94%
1. 7 sides
2. 8 sides
3. 9 sides
6%
0%
si
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7
4. 10 sides
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s
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Can this sequence…
,
,
,
,
help us figure out this sequence?
20
2, 5, 9, 14, _____
Can you find the con nection
What is the link?
1. The numbers tell how
much area the shape has.
2. The shape represents
how many letters are in
the number.
3. The number is a multiple
or factor of the number of
sides of the shape.
4. Something else.
65%
29%
6%
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Hmm?
2,
5,
,
9,
,
20
14, ___
,
,
So what do they have in common?
Challenge #2: Geometry
•
•
•
•
I can make a square with one square.
I can find the perimeter of this square.
I can also find the area.
But what happens to area and
perimeter if I add:
– 1 unit to the length and
– 2 units to the width?
• What if I did the same thing to this
newer, larger rectangle?
• See if you can find a pattern…
Patterns in Architecture
And not just in old
buildings…
Challenge #3
• From nature:
– Try to draw the next set of branches and
leaves.
The number of branches
in the next row is:
1.
2.
3.
4.
5.
6.
7.
8.
12
13
14
15
16
17
18
None of the above
0%
10%0%
10%
10%
70%
12
13
14
15
16
17
18
None of the above
The number of leaves in the
next row is:
7
8
9
10
11
12
13
91%
9%
13
0%
12
0%
11
0%
10
0%
9
8
0%
7
1.
2.
3.
4.
5.
6.
7.
Branches and Leaves
• 21 branches
• 13 leaves
Fibonacci
• Where else can
we find Fibonacci
numbers?
• See if you can
locate Fibonacci
patterns with the
objects given…
http://www.mcs.surrey.ac.uk/Personal/
R.Knott/Fibonacci/fibnat.html
The
th
4
is a fun one!
1 1 2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5 10 3
Can you find the next 3 numbers in the
sequence.
Hmm, there’s something unique about this
pattern.
Check this out!
1 1 2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5 10 3
1
1
2
1
3
2
4
1
5
3
Eliminate every number the first time it
occurs and the result is…
It’s the same pattern! Whoa!
Fractal Pattern
• A pattern within a pattern
within a pattern
• Here are some pictures of
patterns within patterns
Sierpinski’s Triangle
No matter how much you
zoom, you still get the
same picture.
Fractals are natural…
…and man-made.
Our fifth challenge
• Look at the following sequence.
(it uses 1’s and 2’s only).
• Can you find the next 5 numbers?
12211212212…
I’ll let you look at it for a while before I give
you a hint 
Hint: Check out the length of
repetitions of numbers…
Kolakoski Sequence
2 questions to answer:
What should the
next group of
numbers be?
1, 2, 2, 1, 1, 2, 1, 2, 2,
How many
numbers should
the next group
have?
Did we crack
the code?
[email protected]
Thank You !
Extra Slides
Kolakoski Sequence
• Some questions without answers:
– Is there a formula for the nth term?
– If a string (e.g. 21221) occurs in the sequence
will it occur again?
– Why are hotdogs sold in packs of 10 and
buns in packs of 8?
Kolakoski Sequence
2 questions to answer:
What should the
next group of
numbers be?
1, 2, 2, 1, 1, 2, 1, 2, 2,
How many
numbers should
the next group
have?