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# There are n cards in a pile. The cards are numbered 1 to n and they in order with card n on the bottom of the pile and card 1 on the top. Problem: To move all the cards to another pile using the least number of moves.

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Introduction

Investigation: Cards (Algebra)

There are n cards in a pile. The cards are numbered 1 to n and they in order with card n on the bottom of the pile and card 1 on the top.

Problem: To move all the cards to another pile using the least number of moves.

Rules:               1)   Only one card may be moved at a time.

1. The card may only be placed on one of the 3 piles.
2. A larger numbered card may not be placed on a smaller numbered card.

Objective: To find a formula which gives the minimum number of moves for the situation above for 3                       piles. To find a formula which gives the minimum number of moves for 4 piles.

1) Investigation of Problem for 3 piles for small numbers, i.e. from n = 1 to n = 5.

 n Pile 1 Pile 2 Pile 3 No. of moves Total no. of moves No. of card(s) Card number(s) No. of card(s) Card number(s) No. of card(s) Card number(s) 1 1 1 0 - 0 - - 1 = 20 0 - 1 1 0 - 1 2 2 1,2 0 - 0 - - 2 = 21 0 - 1 1 1 2 2 3 3 1,2,3 0 - 0 - - 4 = 22 1 3 1 1 1 2 2 0 - 1 3 2 1,2 2 4 4 1,2,3,4 0 - 0 - - 8 = 23 1 4 1 3 2 1,2 4 2 1,4 2 2,3 0 - 2 0 - 3 1,2,3 1 4 2 5 5 1,2,3,4,5 0 - 0 - - 16 = 24 1 5 3 1,2,3 1 4 8 2 2,5 1 3 2 1,4 2 3 1,2,5 0 - 2 3,4 2 1 5 1 1 3 2,3,4 2 0 - 1 5 4 1,2,3,4 2

Middle

8

5

1,2,3,4,5

1

6

1

7

2

0

-

5

1,2,3,4,5

1

8

2

6,7

2

9

9

1,2,3,4,5,6,7,8,9

0

-

0

-

0

-

-

21

1

9

5

1,2,3,4,5

1

8

2

6,7

17

2

6,9

5

1,2,3,4,5

2

7,8

0

-

2

0

-

5

1,2,3,4,5

3

6,7,8

1

9

2

10

10

1,2,3,4,5,6,7,8,9,10

0

-

0

-

0

-

0

n

Pile 1

Pile 2

Pile 3

Pile 4

No. of moves

Total no. of moves

No. of card(s)

Card number (s)

No. of card(s)

Card number (s)

No. of card(s)

Card number (s)

No. of card(s)

Card number (s)

10

4

7,8,9,10

3

1,2,3

1

6

2

4,5

9

25

5

4,7,8,9, 10

3

1,2,3

2

5,6

0

-

2

4

7,8,9,10

2

2,3

3

4,5,6

1

1

2

5

2,7,8,9, 10

0

-

4

3,4,5,6

1

1

2

4

7,8,9,10

0

-

6

1,2,3,4,5,6

0

-

2

2

9,10

1

7

6

1,2,3,4,5,6

1

8

2

1

10

1

9

6

1,2,3,4,5,6

2

7,8

2

2

7,10

2

8,9

6

1,2,3,4,5,6

0

-

2

0

-

3

7,8,9

6

1,2,3,4,5,6

1

10

2

Table 3

Table 3 shows the least number of moves required for the stated problem for 4 piles.

The table below shows the results found, in short.

 n (least) No. of moves Difference in no. of moves 1 1 - 2 2 1 = 20 3 3 1 = 20 4 5 2 = 21 5 7 2 = 21 6 9 2 = 21 7 13 4 = 22 8 17 4 = 22 9 21 4 = 22 10 25 4 = 22

Table 4

5) Formation of hypothesis for n = 11 to n = 15.

From the values found and shown in Table 4, a hypothesis can be made that for n = 11 to n = 15, there would be an increment of 23 = 8 each time.

Expected values are shown below, using Excel. Table 5

6) Investigation of Problem for 4 piles for small numbers, i.e. from n = 11 to n = 15.

 n Pile 1 Pile 2 Pile 3 Pile 4 No. of moves Total no. of moves No. of card(s) Card number (s) No. of card(s) Card number (s) No. of card(s) Card number (s)

Conclusion

 n Hypothesis Actual 11 33 33 12 41 41 13 49 49 14 57 57 15 65 65

Table 7

7) Deriving of general statement for least number of moves for 4 piles.

Hypothesis is correct.

Therefore, the least number of moves, for the Problem, for 4 piles, increases by powers of 2 that are repeated 2 more than the power, for n that is greater than 1.

When n = 1, the least number of moves is 1.

When n = 2, the least number of moves is 1 + 1(20).

When n = 3, the least number of moves is 1 + 2( 20).

When n = 4, the least number of moves is 1 + 2( 20) + 1(21).

When n = 5, the least number of moves is 1 + 2( 20) + 2(21).

When n = 6, the least number of moves is 1 + 2( 20) + 3(21).

When n = 7, the least number of moves is 1 + 2( 20) + 3(21) + 1(22).

When n = 8, the least number of moves is 1 + 2( 20) + 3(21) + 2(22).

When n = 9, the least number of moves is 1 + 2( 20) + 3(21) + 3(22).

When n = 10, the least number of moves is 1 + 2( 20) + 3(21) + 4(22).

When n = 11, the least number of moves is 1 + 2( 20) + 3(21) + 4(22) + 1(23), so on and so forth.

The numbers that are not inside the brackets must always have a sum of n.

When n = 15, the least number of moves is 1 + 2( 20) + 3(21) + 4(22) + 5(23) = 65, as investigated.

8) Usage of general statement for 4 piles.

Using Excel, the predicted results for n = 16 to n = 25 are as follows. Table 8

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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