Download the activity from the Texas Instruments Australia website.

This challenge involves getting three discs out of a specific area of a checkers board. The region the discs need to escape consists of the six shaded squares in the bottom left corner of the board.

Rule: 1 Discs can only move forward (top of screen) one space at a time.

Rule: 2 Each time a disc is moved it divides into two; one piece going forward and the other immediately to the right.

Rule: 3 Discs cannot be placed on top of one another.

Moves: 0

Moves: 1

Moves: 2

Moves: 3

Moves: 4

Moves: 5

Moves: 6 Only six moves in and one disc has already been cleared from the shaded region. Need to make some room to clear out the next disc.

Moves: 7

Moves: 8

Moves: 9

Moves: 10

Moves: 11

Moves: 12

Moves: 13

Moves: 14

Moves: 15

Moves: 16

Moves: 17 Almost there … How many moves will it take to leave just one piece in the grey region?

Moves: 17 So the board is getting crowded. To make the problem easier, let’s extend the board indefinitely upwards and to the right. How many moves to clear the grey region of all the discs?

1 1 1

Solution Process Let’s say that each of the original discs is given a point value.

1 When a disc is moved and splits into two … so too does it’s value.

1 When a disc is moved and splits into two … so too does it’s value.

1 Solution Process We can see a pattern starting to form.

1 Solution Process Look at the rows

1 Solution Process Look at the rows and the diagonals.

1 Solution Process Look at the rows and the diagonals. Let’s look at the entire board...

Solution Process What is the sum of all the spaces on the board? What if the board were extended indefinitely upward and to the right? 1

Let’s look at the sum of the values in the first row… 1 extended indefinitely.

1

1 And the second row…

1 And the third …

What is the next sum? What is the total sum? 1

1

1 The row sums form a pattern …

1 The sum of the spaces inside the grey area is larger than the sum of the spaces outside. What does this say about the solution?

1 What if the grey region was restricted to just the original discs?

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