Batch 2 - Class 77 - Casting out Nines

• Arrange numbers 1-32 in a circle so that any two adjacent numbers sum to a square
• Answer: 

  

• You are creating a new coin system for a country, and can use at most 4 coin values. You should then be able to create any values from 1 to 10, using up to 2 coins. Create a coin system.
• After you create a successful coin system, other countries ask you create one for them. However, they do not want the system to be repeated. How many different systems can you create?
• Answers: 1237, 1245, 1257, 1258, 1345, 1346, 1356, 1349
• How do you know you have arrived at all the solutions?

• "Digital root" of a number is the result you get if you add up its digits, and then add up the digits of that result, and so on, till you get to a single digit. 
• Illustrate digital root of 44689 - Answer: 4
• Lets look at two numbers that add up to 44689 - such as 31847 and 12842 - what is the relationship between the digital roots of these numbers?
• What about two numbers that subtract to make 44689, like 83491 and 38802? Is there a relationship between their digital roots? What can you do with 100000 and 55311?
• What about two numbers that multiply to make 44689, like 67 and 667? or 23 and 1943?
• The process of taking the digital roots is called "Casting out Nines" - why? Do we really cast away all nines?

• One use of casting is to check arithmetic quickly. If the digital root doesn't match up, then there was an arithmetic mistake. Check the following? Are the other ones actually correct?
• 1234+5678=6812
• 12345-9876=2469
• 10101-2468=7623
• 1234 * 5678 = 7006652
• 4321 * 8765 = 37783565
• 345 * 543 = 196335
• 2^17 = 130072 (How should we handle exponents?)
• Some kinds of mistakes will never be found by casting out nines. Give some examples
• Does this work for division? How can we rewrite division as multiplication and addition, so that we could check by casting out nines? For example: 23894 divided by 82 is 291 with a remainder of 32?
• What is the digital root of 3726125? Is 3726125 a perfect square?
• Lets do a calculator trick - lets pass around the calculator and every kid will press a random digit and multiply key. Once we have done that 25 times, there will be a number on the screen. Another kid will now eliminate one digit (not a 0), and yet another kid will jumble up the digits and write them. A mathematician can tell the missing digit. How?
• Answer: Mathematician assume the digital root of final number is 9, which will be the case if anyone had pressed a 9, or two people had pressed 3 or 6. This can go wrong in some cases though.
• Calculate the digital root of 4444^4444
• Answer: 7. First we can reduce 4444 to 7. Then notice that the digital root cycles every 3 powers of 7.

• Count to 31 - People take turns calling out numbers 1-6, and maintain a cumulative total. The one who gets to 31 first wins. Is there a winning strategy? If so, what?
• Answer: 3, 10, 17, 24, 31 are winning numbers so stay on that - first person wins
• What does this have to do with digital roots?

• Instead of casting out nines, you can also cast out 11s. Start with the rightmost digit, and alternately add and subtract. So with 44689, you'd take 9-8+6-4+4=7. If you land up with a negative number, just add 11 as many times as required. Why does this process work, and can you use this to catch mistakes that you can't catch with casting out 9's?