\n ## Mathematical Problem Solutions: Growing Spirals ### Overview The image presents two identical problem statements regarding growing spirals in the plane, each with its corresponding solution. Both problems ask for the number of points with integer coordinates that cannot be the last point of a growing spiral. The problems differ only in the upper bounds of the coordinate ranges. The solutions are presented step-by-step, culminating in a final answer for each problem. ### Components/Axes The image is divided into two columns, each containing a problem statement and its solution. Each column is further divided into sections: 1. **Problem Statement:** Defines a growing spiral and poses a question about the number of points that cannot be the last point of the spiral. 2. **Solution:** Provides a step-by-step explanation of how to determine the number of excluded points. 3. **Final Answer:** States the calculated number of excluded points. Each column also includes a "Year: 2011 ID: A1" label at the bottom. ### Detailed Analysis or Content Details **Left Column:** * **Problem:** Defines a growing spiral with points (x, y) where 0 ≤ x ≤ 2011 and 0 ≤ y ≤ 2011. * **Solution:** * Points that cannot be the last point are (0, y) for 0 ≤ y ≤ 2011, (x, 0) for 1 ≤ x ≤ 2011, (x, 2) for 2 ≤ x ≤ 2011, and (x, 3) for 3 ≤ x ≤ 2011. * Total excluded points calculation: 2012 + 2011 + 2011 = 6034. Further calculation: +2010 + 2009 = 10053. * **Final Answer:** 10053 **Right Column:** * **Problem:** Defines a growing spiral with points (w, v) where 0 ≤ w ≤ 4680 and 0 ≤ v ≤ 4680. * **Solution:** * Points that cannot be the last point are (0, v) for 0 ≤ v ≤ 4680, (w, 0) for 1 ≤ w ≤ 4680, (w, 2) for 2 ≤ w ≤ 4680, and (w, 3) for 3 ≤ w ≤ 4680. * Total excluded points calculation: 4681 + 4680 + 4680 = 14041. Further calculation: +4679 + 4678 = 23398. * **Final Answer:** 23398 ### Key Observations * The problems are structurally identical, differing only in the upper bounds of the coordinate ranges (2011 vs. 4680). * The solution approach is the same for both problems. * The final answers are significantly different, reflecting the difference in the coordinate ranges. * The calculations are presented in a step-by-step manner, making the logic clear. ### Interpretation The problems demonstrate a mathematical concept related to growing spirals and the exclusion of certain points as potential endpoints. The solutions involve identifying the points that cannot be the last point of the spiral based on the defined constraints and then calculating the total number of such points. The difference in the final answers between the two problems highlights the impact of the coordinate range on the number of excluded points. The problems are likely designed to test a student's understanding of coordinate geometry, sequences, and problem-solving skills. The "Year: 2011 ID: A1" suggests these are problems from a 2011 assessment.