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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% it ith w s cl as ny A m at h Cl a rts A e ua g ng La in ss s la s C s C So ci al St ud ie ce ie n Sc at h C la s la s s s 0% 0% 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% w ne A A ov e. ab ft he ll o rr un ... ... co sy ia l ec sp A m pu te st em n tte r pa a in g ak us e w hi c. rs m be nu up ng M M .. or ... 0% ix i 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 de 10 s 9 si de s si de 8 7 4. 10 sides si de s s 0% 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% ... . el ng m et hi s ri m be nu e Th So m a re s re p e ap sh e Th ul ti en t.. .. ho . te ll rs m be nu e Th se . 0% 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?