Let's carefully analyze the problem step-by-step. --- ### **Given Data:** - **Varieties:** 0, 1, 2, ..., 10 (total 11 varieties) - **Starter block:** {1, 4, 6, 7, 8} - **Design:** Cyclic design (implies the blocks are generated by cyclically shifting the starter block) --- ### **Part 1: Construct the difference table for the starter block** **Objective:** Find all pairwise differences modulo 11 (since the varieties are 0 to 10). --- ### **Step 1: List the starter block** \[ B = \{1, 4, 6, 7, 8\} \] ### **Step 2: Compute differences between each pair** - Differences are computed as \( (b_j - b_i) \mod 11 \), for \( i < j \). --- ### **Step 3: List all pairs and their differences** | Pair | Difference \( (b_j - b_i) \) | Modulo 11 | Result | |---------|------------------------------|-----------|---------| | (1,4) | 4 - 1 | 3 | 3 | | (1,6) | 6 - 1 | 5 | 5 | | (1,7) | 7 - 1 | 6 | 6 | | (1,8) | 8 - 1 | 7 | 7 | | (4,6) | 6 - 4 | 2 | 2 | | (4,7) | 7 - 4 | 3 | 3 | | (4,8) | 8 - 4 | 4 | 4 | | (6,7) | 7 - 6 | 1 | 1 | | (6,8) | 8 - 6 | 2 | 2 | | (7,8) | 8 - 7 | 1 | 1 | ### **Step 4: Summarize the difference multiset** | Difference | Count | |--------------|--------| | 1 | 2 | | 2 | 2 | | 3 | 2 | | 4 | 1 | | 5 | 1 | | 6 | 1 | | 7 | 1 | --- ### **Part 2: Construct the difference table** The difference table is typically a symmetric matrix showing the differences between all pairs (excluding the diagonal). Since the differences are between pairs, and the counts are as above, the difference table can be visualized as: | | 1 | 4 | 6 | 7 | 8 | |-------|---|---|---|---|---| | **1** | - | 3 | 5 | 6 | 7 | | **4** | | - | 2 | 3 | 4 | | **6** | | | - | 1 | 2 | | **7** | | | | - | 1 | | **8** | | | | | - | *(Note: The lower triangle is symmetric, and the diagonals are ignored.)* --- ### **Part 3: Write down the first row of the concurrence matrix** The **concurrence matrix** \( C \) is an incidence matrix indicating whether pairs of varieties co-occur in blocks. - For the **initial block** {1, 4, 6, 7, 8}, the first row corresponds to variety 1. - The **first row** of the concurrence matrix will have 1s in positions corresponding to varieties that co-occur with 1, i.e., 4, 6, 7, 8, and 0 elsewhere. \[ \boxed{ \text{Row for var. 1:} \quad [0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0] } \] *(Assuming the varieties are ordered 0 to 10).* --- ### **Part 4: Determine whether the design is balanced** - **Balanced Incomplete Block Design (BIBD):** A design where **each pair** of varieties occurs together in exactly \( \lambda \) blocks. - **In our case:** - The **difference multiset** shows differences occur with counts 2, 2, 2 for differences 1, 2, 3, and 1 each for others. - Since the pairwise differences are not uniform (some differences occur more frequently), **the design is *not* balanced**. --- ### **Part 5: Is the starter block a perfect difference set?** - **A difference set** is a subset \( D \) of a group \( G \) such that every non-zero element of \( G \) can be expressed as a difference of two elements of \( D \) exactly \( \lambda \) times. - Here, the multiset of differences shows **some differences occur multiple times**, not uniformly, so **it is not a perfect difference set**. --- ### **Part 6: Without constructing any designs, determine whether the complement of the block forms a balanced incomplete block design** - **Complement of the starter block:** \( \overline{B} = \{0, 2, 3, 5, 9, 10\} \) - For the complement to form a BIBD or be balanced, the frequencies of pairwise incidences should be uniform. - **Given the initial differences are not uniform, the complement is unlikely to form a balanced incomplete block design.** --- ### **Summary & Conclusions** | **Question** | **Answer** | **Explanation** | |--------------|--------------|-----------------| | Constructed difference table? | Yes | List of all pairwise differences modulo 11. | | First row of concurrence matrix? | [0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0] | Based on initial block co-occurrences. | | Is the design balanced? | **No** | Differences are not uniformly distributed; pairs do not occur equally. | | Is the starter block a perfect difference set? | **No** | Differences occur with different frequencies; not uniform. | | Is the complement a BIBD? | **No** | Differences indicate imbalance; unlikely to form a BIBD. | --- ### **Final Note:** This analysis shows that while the cyclic design generated from the starter block provides a systematic way to generate blocks, the initial block's difference properties determine the balance and difference set qualities of the design. --- **If you need further clarification or detailed calculations, feel free to ask!**