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Similarly, if you want to represent thirds, you should choose the shortest rod that has a three-car, one-color train and name that rod "1."
The shortest rod that has a three-car, one-color train is the light green rod. If light green is "1," then white is 1/3, and we could name all other rods in terms of these two rods: |
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The next-longest rod that has a three-car, one-color train is the dark-green rod. It has a three-car all-red train, as well as a light green, and a white train. Notice that the thirds in this case are red rods. If we name the dark-green rod 1, then the light-green rod is 1/2, the red rod is 1/3, and the white rod is 1/6. |
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The next-longest rod to satisfy the requirement is the blue rod, which has a three-car, light-green train, and also a white train: |
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In fact, we can show that every rod that has a three-car, one-color train also has a light-green train, so any time we want to deal with thirds, we must choose a rod with an all-light-green train to represent "1."
Consequently, if we want to deal with halves and thirds at the same time, we need a rod that has both an all-red train and an all-light-green train. As we've seen above, this is the dark-green rod. If we call the dark-green rod "1," then white is 1/6, red is 1/3, and light green is 1/2, and all other rods can be named in terms of these rods.
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