Lesson plan
Sparse ruler
A sparse ruler is a ruler in which some of the distance marks may be missing. More abstractly, a sparse ruler of length L with m marks is a sequence of integers a_1, a_2, ..., a_m where 0 = a_1 . The marks a_1 and a_m correspond to the ends of the ruler. In order to measure the distance K , with 0\le K\le L there must be marks a_i and a_j such that a_j-a_i=K . A complete sparse ruler allows one to measure any integer distance up to its full length. A complete sparse ruler is called minimal if there is no complete sparse ruler of length L with m-1 marks. In other words, if any of the marks is removed one can no longer measure all of the distances, even if the marks could be rearranged. A complete sparse ruler is called maximal if there is no complete sparse ruler of greater length with m marks. Complete minimal rulers of length 135 and 136 require one more mark than those of lengths 124-134, 137 and 138. A sparse ruler is called optimal if it is both minimal and maximal. Since the number of distinct pairs of marks is m(m-1)/2 , this is an upper bound on the length L of any maximal sparse ruler with m marks. This upper bound can be achieved only for 2, 3 or 4 marks. For larger numbers of marks, the difference between the optimal length and the bound grows gradually, and unevenly. For example, for 6 marks the upper bound is 15, but the maximal length is 13.…
Article summary
A sparse ruler is a ruler in which some of the distance marks may be missing. More abstractly, a sparse ruler of length L with m marks is a sequence of integers a_1, a_2, ..., a_m where 0 = a_1 . The marks a_1 and a_m correspond to the ends of the ruler. In order to measure the distance K , with 0\le K\le L there must be marks a_i and a_j such that a_j-a_i=K . A complete sparse ruler allows one to measure any integer distance up to its full length. A complete sparse ruler is called minimal if there is no complete sparse ruler of length L with m-1 marks. In other words, if any of the marks is removed one can no longer measure all of the distances, even if the marks could be rearranged. A complete sparse ruler is called maximal if there is no complete sparse ruler of greater length with m marks. Complete minimal rulers of length 135 and 136 require one more mark than those of lengths 124-134, 137 and 138. A sparse ruler is called optimal if it is both minimal and maximal. Since the number of distinct pairs of marks is m(m-1)/2 , this is an upper bound on the length L of any maximal sparse ruler with m marks. This upper bound can be achieved only for 2, 3 or 4 marks. For larger numbers of marks, the difference between the optimal length and the bound grows gradually, and unevenly. For example, for 6 marks the upper bound is 15, but the maximal length is 13.…
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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