| Original Problem: | My neighbor has a peculiar orchard. He has five apple trees, five peach trees, five pear trees, five apricot trees, and five plum trees. The trees are planted in a square of five rows of five trees each. Each row (in each direction) has exactly one tree of each variety and so do the center diagonals. How are the trees planted? |
This problem is called Planting The Orchard. The orchard has 5 apple trees, 5 plum trees, 5 peach trees, 5 pear trees and 5 apricot trees. I'm supposed to tell how to plant the trees in 5 rows so that there no two trees in a row are the same kind.
2. Process
To prepare to solve the problem the first thing I did was to make a square with five rows with five squares in each row. Next I assigned a number to each tree.
| 1 = apple |
2 = plum |
3 = peach |
4 = pear |
5 = apricot |
and I cut out 25 small squares with each number on five squares. My plan was to use guess and check to try all the possible ways of planting the trees. I thought would put the tree squares on the orchard square in different ways to see if I could solve the puzzle.
To start out, I filled in one row of trees in order, 1 2 3 4 5. That was the simplest thing I could try. Then I tried putting trees down lots of ways but nothing worked even after a long time. Then I got an idea. I tried to put down all of the squares with 1s on them first. And I used process of elimination to help me. Each time I planted a tree I crossed off all the squares that couldn't have that kind of tree in it anymore.
Example
My orchard started out like this:
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Since there is already a 1 in the corner I couldn't plant a 1 in the first column or in the diagonal. I marked these places with Xs to show this.
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To make the puzzle work there had to be a 1 in every row and diagonal. There had to be a 1 in the second row so I put it in the first spot I could, under the 3. Then I put Xs in the other squares in that row and column. Now I had
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Now there was only one open square on the other diagonal so a 1 had to go there. Crossing out the other squares again I had
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Now there was only one spot left in the last column so I put a 1 there and crossed out the row and column. That left only one empty square which I filled in with the last 1. Since I was done with the 1s I erased the Xs and got this
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I did the same thing with the other numbers. It took a long time but it finally worked!
The Answer
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3. Evaluation
I felt that this problem was very hard. It took a long time and many tries to get the answer. I liked it because I like logic puzzles and part of it was like a logic puzzle. The hardest part was going through all the possible choices in order. I forgot sometimes what I had tried and what I hadn't. I learned that guessing and checking takes a long time and sometimes logic can help speed things up. I'm proud that I finally solved this problem.
After I finished my Dad showed me a pattern in the answer. Every two trees that are the same are connected by an "L" shape like this.
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If only I had known that before I started I could have been done in 5 minutes!