Batch 3 - Class 217 - Marching Squares, Ariadne's String

• Research effects of small biases leading to large difference in outcomes in different areas. Research initiatives that are underway to correct those outcomes.

• Discuss marching patterns with kids
• Normal march forward - 8 to 5 steps - what does each marcher have to do?
• More complex patterns, like a rotating square (4 marchers) - what would you direct them to do?
• We want to create a marching pattern, where two sets of marchers, each in an equal square grid pattern come together and create a larger square. So for example, two 5x5 groups come together to form a larger square unit
• Let kids try to figure out a set of numbers that work
• If they are not able to get to a number is a bit, they may say its not possible
• Can they prove it?
• Let us assume that there is some "smallest" number for which this is possible
• What will happen when those two merge?

• What is the implication of this? What must the red square equal up to?
• What does that mean about our assumption of the initial squares being "smallest" such squares?
• What can we conclude? There is no smallest square, and hence no such square possible!
• Can you prove this algebraically? We can use this to prove that square root ot 2 is irrational.
• Note that in this case, we can get "near misses" - such as two 12 squares getting us to 17 square with 1 miss. We can now start to scale this down to smaller "near misses". We can also scale these up to larger "near misses". Find a few of these. What is the relationship between these?
• What if there was no requirement for these squares to be equal?

• You are guiding Theseus through a room full of pillars. Theseus starts by tying his string to a column on the perimeter of the room. He then moves around tying it to another pillar, always increasing the distance, and ensuring that the string doesn't intersect itself. How many segments can Theseus create?
• Start with a 2x2 pillar grid
• Try a 5x5 pillar grid

• If you reach the minotaur too soon gets gored. If you do well, but not optimal, you kill the minotaur but are unable to escape. The optimal solution kills the minotaur and Theseus can escape. 
• The optimal solution for 5x5 grid is 9 segments

• Try larger grid sizes
• Some of the optimal solutions here https://static01.nyt.com/packages/pdf/crossword/Ariadne_String_Grids_2x2_to_9x9_Solutions.pdf

• A marching band is having trouble lining up. But
• When they line up in 2's, there's 1 left over
• When they line up in 3's, there's 2 left over
• When they line up in 4's, there's 3 left over
• When they line up in 5's, there's 4 left over
• When they line up in 6's, there's 5 left over
• When they line up in 7's, there's none left over
• How many are there? Explore patterns due to 7 being relatively prime to all other numbers. What if it wasn't - suppose we extend this to 8?