Batch 3 - Class 135 - Miscellaneous, Look and Say Sequence

• There are three printing machines. One takes a card (a,b) and returns a card (a+1,b+1). The second takes a card with even numbers on it and returns (a/2,b/2). The third takes two cards (a,b) and (b,c), and returns (a,c). Each machine also returns back the initial cards. Starting from (5,19), can you get to (1,1988)
• Work through steps with kids

  In first machine, what could be the invariant? Difference of the two numbers. In the second, the difference gets halved. In the third, the differences get added. So the difference is not the invariant, but the invariant may have to do something with the difference

  Now lets observe for few steps

  (5,19) -> (6,20)(6,20) -> (3,10)(3,10) -> (4,11) ... -> (20,27)(6, 20) and (20,27) -> (6,27)What are the differences - 14, 14, 7, 7, 21 - All divisible by 7! Can we now prove the invariant?
• Since 1988-1 is not divisible by 7, we can't get that card

• Connect the small boxes below with their counterparts, without crossing any lines or leaving the box

• Answer: 

• A monk climbs a mountain. He starts at 8AM and reaches the summit at noon. He spends the night on the summit. The next morning, he leaves the summit at 8AM and descends by the same route that he used the day before, reaching the bottom at noon. Prove that there is a time between 8AM and noon at which the monk was at exactly the same spot on the mountain on both days.
• Answer: Let the monk climb up whichever way he wants. At the instant he begins his descent the next morning, have another monk hike up traveling exactly as the first monk did the first day. At some point, the two monks will meet on the trail. That is the time and place we want!
• What's the next term in the sequence 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211 ...
• Answer: 31131211131221 - This is called the Look and Say Sequence, because the next term in the sequence is just the way you would pronounce the previous term. 
• If we start from any difit from 0 to 9, then the last digit always remains d. In particular, the sequence will be d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d and so on. For d=3, this is called the Conway sequence

  Note that apart from d, this sequence only has digits 1, 2 and 3This is also true of any larger starting numbers, as long as it doesn't have a run of length > 3 to start with
• Normally, this sequence grows indefinitely - can you find a sequence which will not grow?

  Answer: 22, 22, 22, ...The rate at which any sequence can grow from one term to another is limited to around 30%. It means that (Length of term n+1/Length of term n) is about 1.3 as n grows large. The very fact that this kind of a limit exists is surprising.
• Another pattern "pea pattern" entails taking a term and listing number of total occurrences of each digit in the order they appear. So 1, 11, 21, 1211, 3112, 132112, 311322

  This version eventually makes a cycle 23322114, 32232114Any pea pattern eventually produces terms of bounded length, and hence cycles
• How many times must you flip a fair coin, to have at least 50% probability of getting at least 3 heads?
• Answer: 5. P (at least 3 heads)+P (at least 3 tails) = 100%. Heads and Tails are symmetric

• Of all the 10000 books at the library, if you select one at random, then there is a 90% chance that it has illustrations. Of all the illustrations in all books, if you select one at random, then there is a 90% chance that it is in color. What is the minimum number of books that must contain colored illustrations?
• Answer: 1. One book can have a lot of colored illustrations, and all other 8999 books can have one uncolored one.