Batch 3 - Class 239 - Blue Dot Solitaire

• Why is angle of incidence the same as angle of reflection?
• Light is not really particulate - in quantum theory, "who can see whom"?

• Make  a line of blue and yellow dots. At any stage you can remove a blue dot - when you do so, the dots adjacent to it change color. Is there a way to remove all dots?
• Example: B Y Y B B Y
• Create and play with your own pattern of dots. Try to answer the following questions:
• What would be a good strategy to remove all dots if you started with only 3 dots?
• How many different patterns can you find with 4 dots? What makes them different? For which of these patterns can you remove all of the dots?
• Start with 1 blue dot and 3 yellow dots. Can you remove all of the dots? How about 1 blue dot and 4 yellow dots? 1 blue dot and any number of yellow dots?
• Come up with your own question(s) and try to answer them. Share them with others!
• Working through
• Ask kids to start with one dot, then two, then three and so on.
• Ask them to list patterns that are unsolvable
• Do they see commonality in those patterns? Can they reason why such commonality should exist? Can they come up with some rules? (ignore symmetrical patterns)
• 1 dot: Y is unsovable
• 2 dot: BB and YY are unsolvable
• 3 dots: BBY, BYB and YYY are unsolvable
• 4 dots: YBBY, BBYY, BYYB, BYBY, BBBB and YYYY are unsolvable
• Some rules that appear
• If you take the smallest subsequence with all Blues of the original sequence part of it, and the subsequence is unsolvable, then the original sequence is unsolvable. For example: Since BYYB is unsolvable, YYBYYB will also be unsolvable. Why? (because anything outside of this subsequence will not affect the solvability of this subsequence itself - the outside yellows will have to turn blue to affect this subsequence, but when they turn blue, they are already severed from this subsequence)
• Even term BBBBBB..B is not solvable. Why? Because regardless of where you start, you are left with a sequence where the largest subsequence is a smaller BBB..B
• BRR..RB is unsolvable. Why? You have to start at an end, and you are left with smaller BR..RB sequence. You can also think of this as finally leading to BB which is unsolvable.
• Any others?
• Can you find an easier mechanism?
• Challenge questions
• Find at least two rows of 6 dots where it is impossible to remove all the dots. Why is it impossible for these patterns?
• Given any row of 6 dots, can you figure out a way to determine whether it is possible to remove all dots just by looking at the initial sequence?
• What if you started with 5 dots? 7 dots? N dots?
• Answer: If there are even number of blue dots, then the pattern is not solvable. If its odd number of dots, the pattern is solvable
• Proof: Suppose for any strings of length upto L and even blues, it is unsolvable (establish base case for L=1,2,3 by exhaustive listing), then for a (L+1), regardless of which blue is taken out, you will have one substring of length <=L and even blues. 
• Similarly, for odd blues, just take the first blue from one end, and take that out. You will now be left with two substrings of lesser length, and both having odd blues, and hence solvable.

• (Mott - 32) Mysterious Computation. Professor Digit claims that 161+134+145=503. Further, he also claims that 2625 when divided by 52 gives a quotient of 36 and zero remainder. Is he nuts?
• Answer: The computations are being made in base 7 instead of base 10