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Fewest Number of Moves Necessary for the Pinch Hitter - Honors Algebra










































The dugout at the Central High School baseball field is shared by both the home team and the visiting team. There is a bench in the dugout for "pinch" hitters. The right side of the bench is designated for the pinch hitters of the team that is at bat. The left side of the bench is designated for the pinch hitters of the team that is in the field. The two teams are separated by one empty seat. When the teams at play change positions, the pinch hitters must also shift positions; the hitters on the left side must move to the right side and the hitters on the right side must move to the left side.
The long awaited game between rivals Central high and Dayton high was on Sunday. There were 4 pinch hitters from central and 6 pinch hitters from Dayton on the bench. Before the game, Coach Madden told the pinch hitters, "Follow my rules carefully so that you can make the switch in the fewest number of moves." What is the fewest number of moves necessary for the pinch hitters to make the switch?
Shifting Rules:
1.) Only one player may move at a time
2.) A player initially on the left side may only move to the right and a player initally on the right side may only move to the left.
3.) A player, on his move, may pass in front of only one player of the OPPOSING team or move into an adjacent empty seat.

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Mar 10, 2012
Fewest Number of Moves Necessary for the Pinch Hitter
by: Staff

Part I
Question:
The dugout at the Central High School baseball field is shared by both the home team and the visiting team. There is a bench in the dugout for "pinch" hitters. The right side of the bench is designated for the pinch hitters of the team that is at bat. The left side of the bench is designated for the pinch hitters of the team that is in the field. The two teams are separated by one empty seat. When the teams at play change positions, the pinch hitters must also shift positions; the hitters on the left side must move to the right side and the hitters on the right side must move to the left side.

The long awaited game between rivals Central high and Dayton high was on Sunday. There were 4 pinch hitters from central and 6 pinch hitters from Dayton on the bench. Before the game, Coach Madden told the pinch hitters, "Follow my rules carefully so that you can make the switch in the fewest number of moves." What is the fewest number of moves necessary for the pinch hitters to make the switch?
Shifting Rules:

1.) Only one player may move at a time
2.) A player initially on the left side may only move to the right and a player initially on the right side may only move to the left.
3.) A player, on his move, may pass in front of only one player of the OPPOSING team or move into an adjacent empty seat.

Answer:

This problem is a variation of the fun game called “Traffic Jam”.

The easiest way to develop a technique which will solve this problem is to use manipulatives. Plastic pieces, coins – anything will do.

Set up 4 pieces on the left side of the empty seat on the bench, and 6 pieces on the right side of the empty seat on the bench.

However, to develop a mathematical relationship, start it with only two pieces – one left, one right.

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Mar 10, 2012
Fewest Number of Moves Necessary for the Pinch Hitter
by: Staff

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Part II

Two People – one left, one right. This takes 3 moves


Start: Central empty Dayton

1st move: Empty Central Dayton
2nd move: Dayton Central empty
3rd move: Dayton empty Central


Four People – two left, two right. This takes 8 moves.

Start: Central Central empty Dayton Dayton

1st move: Central empty Central Dayton Dayton
2nd move: Central Dayton Central empty Dayton
3rd move: Central Dayton Central Dayton empty
4th move: Central Dayton empty Dayton Central
5th move: empty Dayton Central Dayton Central
6th move: Dayton empty Central Dayton Central
7th move: Dayton Dayton Central empty Central
8th move: Dayton Dayton empty Central Central


Six People – three left, three right. This takes 15 moves.
.
.
.





# pairs . . . . . . # people . . . . . # moves . . . . . formula
1 . . . . . . . . . . .2 . . . . . . . . . . .3 . . . . . . . . . . .1² + 2*1 = 3
2 . . . . . . . . . . .4 . . . . . . . . . . .8 . . . . . . . . . . .2² + 2*2 = 8
3 . . . . . . . . . . .6 . . . . . . . . . . .15 . . . . . . . . . .3² + 2*3 = 15
4 . . . . . . . . . . .8 . . . . . . . . . . .24 . . . . . . . . . .4² + 2*4 = 24

n = # of pairs.

If the number of "pinch" hitters for each team is equal, the minimum number of moves

= n^2 + 2(n)

= n(n + 2)



If there are 4 pairs, it takes 24 moves for the pinch hitters to trade places.

However, Dayton has 6 "pinch" hitters – two more than Central.

I’ll let you work this last part out using manipulatives.




Thanks for writing.

Staff
www.solving-math-problems.com


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