Download Summary of the Meeting

NC State Math Circle: October 25, 2014 Prime F actorization a nd I nteger S equences Toothpick Puzzles Directions: Arrange 12 toothpicks in a 2x2 grid: this is your starting position. Puzzle 1: Make two squares of different sizes by removing 2 toothpicks. Puzzle 2: Make three congruent squares (recall we defined “congruent” as “same shape and same size”) by moving 3 toothpicks. Puzzle 3: Make three congruent squares by moving three toothpicks. Puzzle 4: Make seven squares, not all congruent, by moving two toothpicks. Hint: you may cross one toothpick over another! Puzzle 5: Make ten squares by moving 4 toothpicks. Solutions: Puzzle 1 Puzzle 2 Puzzle 3 Puzzle 4 Puzzle 5 Switching Light Bulbs Question: A long hallway has 1000 light bulbs with pull strings, numbered 1 through 1000. If the light bulb is on, then pulling the string will turn it off. If the light bulb is off, then pulling the string will turn it on. Initially, all the bulbs are off. At one end of the hallway, 1000 people numbered 1 through 1000 wait. Each person, when they walk down the hallway, will pull the string of every light bulb whose number is a multiple of theirs. So, for example, person 1 will pull every string; person 2 will pull the strings of bulbs number 2, 4, 6, 8, 10, …. Which bulbs are on after all 1000 people finish walking? Observations: 1. # of times light bulb n gets pulled = # factors/divisors of the number n. 2. If a light bulb is pulled an ODD number of times, then it is ON. 3. If a light bulb is pulled an EVEN number of times, then it is OFF. 4. Prime numbers have only two factors: 1, and themselves. So they will be OFF! 5. In general, the factors of a number n can be grouped into pairs: k x (n/k) = n, so we group the factors k and n/k together. For example, the divisors of 12 are 1, 2, 3, 4, 6, 12. We pair off these factors as {1,12}, {2,6}, {3,4}. If factors can be grouped into pairs, there must be an EVEN number of them. 6. However, if a number n is a perfect square; that is, n = m2 for some other integer m, then the factor m gets paired WITH ITSELF! Solution: There are 31 light bulbs left on after everyone has finished walking, corresponding to the 31 perfect squares between 1 and 1000: 1, 4, 9, 16, 25, 36, 64, 81, 100, 121, 144, 189, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961. Extensions: There are many more questions to think about when it comes to switching light bulbs! See the Switching Light Bulbs worksheet for some more, and try to come up with more interesting problems to consider. Strategies for Problem Solving: 1. Try out a smaller example! Many of you were able to get some good intuition by looking and what happened with the first 10 light bulbs. 2. Look for patterns! If you know for sure that light bulbs n1 and n2 are on while n3 is off, think about what characteristics the numbers n1 and n2 share that n3 does not. 3. Think about the Fundamental Theorem of Arithmetic that I presented: Every positive integer (except 1) can be written uniquely as the product of prime powers. That is, n = p1m1 x p2m2 x ... x pkmk for some prime numbers p1, p2, …, pk and positive integer powers m1, m2, …, mk. For example, 4 = 22, 100 = 22 x 52, 504 = 23 x 32 x 7. Can you use this Theorem to count the number of factors of a given number in a systematic way? The “Switching Light Bulbs” problem is a variation on the “Locker Problem,” and was introduced to me by Car Talk, the NPR show. You can find lots of other good puzzles on the Car Talk website: http://www.cartalk.com/content/puzzlers. Chomp Check out the directions to the game and some intriguing questions here: http://www.math.cornell.edu/~mec/2003-­‐
2004/graphtheory/chomp/howtoplaychomp.html. Let me know if you come up with any cool variations! Crack the Graham If you want to think some more about prime factorization (even if it seems like you aren’t) try your hand at the Crack the Graham worksheet. Warning: this is a challenge! Here is an example of a sequence that satisfies conditions A(6)-­‐E(6): 6, 8, 9, 12, 36. Thanks for coming! Have more questions or ideas? Email Shira at [email protected]!