Move one coin to make two rows of 5

Lecture 7

Today

Midterm

P a r t 1:

W h y

Why we do this

Critical thinking!

Problem solving!

Keys to Mastery

Repetition

Challenge

Define the Problem

don't jump into trying to solve

read

re read

remain open minded

clarify

break it down

refine/be specific

figure out whys of the problem

P a r t 2:

Q u e s t i o n 1

1. You have 20 coins in a row. Ten are heads up, ten are tails up. The order is random. You also have a robot that can perform 4 different actions:

Move itself one coin's width to the left or right at a time.

See the heads/tails status of the two coins directly in front of it.

Flip over both of the two coins that it sees.

Remember a number

Can you give the robot a set of instructions that will result in all 20 coins eventually being turned heads up using only the capabilities outlined above? If not, why? If so, what are the instructions?

Understand

Look for relations

Activity revisited

Can we flip successive pairs to leave only heads facing up?

Activity revisited

Either increase heads by 2

Decrease heads by 2

Or number remains unchanged

Simplify

Simplify

Simplify

Simplify

Simplify

Simplify

Simplify

Simplify

Activity revisited

look at 2 coins in front

if left is a tails, flip both

move one step to the right

repeat from 2

Activity revisited

remember the number 1

look at the 2 coins the robot can see

if left is a tails, flip both

move one step to the right

increase the number remembered by one

if only one coin can be seen in front of robot:

If number matches number of coins, Finish

otherwise: take one step to the left

swap all instances of left and right

change number remembered to 1

repeat from 3

Number: 1

Number: 1

Number: 2

Number: 2

Number: 3

Number: 3

Number: 4

Number: 5

Number: 6

Number: 6

Number: 6

Number: 6

Number: 1

Number: 1

Number: 2

Number: 2

Number: 3

Number: 3

Number: 4

Number: 4

Number: 5

Number: 6

Number: 7

Number: 8

finish

testing a solution

clarifications

edge cases

P a r t 3:

Q u e s t i o n 2

move space one to left

move space one to right

move space two to left

move space two to right

slide left facing coin left

slide right facing coin right

jump left facing coin left

jump right facing coin right

look for patterns

2

explore pattern

start small

keep trying

look for patterns

look for patterns

start direction left

move space in direction by sliding

make as many jumps over opposite pieces going direction as possible

switch direction

repeat from step 2 until can no longer move space, then switch to:

switch direction

move space in direction by sliding

make as many jumps over opposite pieces going opposite direction as possible

repeat from step 6

1. start direction left

2. move space in direction by sliding

3. make as many jumps over opposite pieces going direction as possible

4. switch direction

5. repeat from step 2 until can no longer move space

2. move space in direction by sliding

3. make as many jumps over opposite pieces going direction as possible

4. switch direction

5. repeat from step 2 until can no longer move space

2. move space in direction by sliding

3. make as many jumps over opposite pieces going direction as possible

4. switch direction

5. repeat from step 2 until can no longer move space

2. move space in direction by sliding

3. make as many jumps over opposite pieces going direction as possible

4. switch direction

5. repeat from step 2 until can no longer move space

then switch to:

6. switch direction

7. move space in direction by sliding

8. make as many jumps over opposite pieces going opposite direction as possible

9. repeat from step 6

6. switch direction

7. move space in direction by sliding

8. make as many jumps over opposite pieces going opposite direction as possible

6. switch direction

7. move space in direction by sliding

8. make as many jumps over opposite pieces going opposite direction as possible

6. switch direction

7. move space in direction by sliding

8. make as many jumps over opposite pieces going opposite direction as possible

6. Switch direction

7. move space in direction by sliding

explore pattern

start small

keep trying

P a r t 4:

Q u e s t i o n 3

3. You and a friend are making a tower out of blocks. You take turns each adding between one and three blocks to the tower. You both know that as soon as the tower is 30 blocks high, it will topple. Each person's goal is to avoid being the one who topples the tower.

The first person to place a block has a strategy that will guarantee they can avoid being the one who topples the tower. What is it and why does it work?

simplify

Tower topples at 1

simplify

Tower topples at 1

simplify

Tower topples at 1

Player 1 loses

simplify

Tower topples at 2

simplify

Tower topples at 2

simplify

Tower topples at 2

Player 2 loses

simplify

Tower topples at 3

simplify

Tower topples at 3

simplify

Tower topples at 3

Player 2 loses

simplify

Tower topples at 4

Player 2 loses

generalize: you win if you start your turn 2, 3, or 4 blocks away from finish

how can we guarantee that?

what can we guarantee each round?

2-6 is the total bocks possible each round, but we can't guarantee all of those results

we can guarantee 4 though!

how?

we enter our first scenario when the opponent has their turn 5 blocks from tower topple

by locking in 4 each round we win then if the other player goes with 9 blocks until topple

which we can guarantee from 13

extended to:

17

21

25

29

so we want to have the other player have their turn with 29 blocks remaining until topple

which we can guarantee by adding one block when we go first

then adding 4 - (however many blocks the other player just added) each turn

extending a pattern

test assumptions

try to break your solutions

4. Ask me a question.

Questions

what do you want to know

Questions

question's purpose

(why do you want to know it)

Questions

intent

broader reasons

Questions

phrasing the question itself

Questions

follow through

P a r t 5:

Q u e s t i o n 4

Example Questions

Why...

How does...

What caused...

What is meant by...

What is the connection between...

How is this like..

Example Questions

How do you start your approach

Example Questions

When do I know to stop?

Example Questions

Why did I decide to teach this course?

Example Questions

Are some people naturally gifted problem solvers?

Example Questions

Where did you get idea to do end of class journal?

Using only four lines with continuous strokes are you able to connect all the dots?

Journal

1 2 3 4 5 6 7

2. There are 6 coins on 7 s paces arranged as above. Coins c an only be

moved one at a time to an empty space. A coin can jump over at most one

other coin at a time. Coins can only move in the direc tion of their arrow.

No rotating coins.

What sequence of moves will result in all of the left pointing coins ending

up in positions 1, 2, and 3 while the right pointing coins end up in

positions 5, 6, and 7?

Can you generalize a pattern of movement that would enable us to

extend the number of coins and spaces to an arbitrary amount (say 20

coins with 21 spaces) without drastically increasing the number of

written steps?