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.
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.
- The card may only be placed on one of the 3 piles.
- 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.