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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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.

Table 1

2) Deriving of formula for least number of moves for 3 piles.

As shown in Table 1, the least number of moves is always a power of 2. Or more precisely, (n-1)th power of 2.

Therefore, the least number of moves for the Problem for 3 piles can be calculated using the formula

2(n-1),

n being the number of cards from 1 to n.

3) Usage of formula for 3 piles.

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Using Excel, the predicted results for n=6 to n=15 are as follows.

Table 2

The number of moves was calculated using the fill down option. The column showing the formula in Table 2 shows the function, power, used to calculate in Excel.

4) Investigation of the Problem for 4 piles for small numbers, i.e. from n = 1 to n = 10.

Table 3

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

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