$ q_3 = p_3 - 2 \le 8 - 2 = 6 $, but with $ q_1 \ge 1 $, $ q_2 \ge q_1 + 1 $, $ q_3 \ge q_2 + 1 $, and $ q_3 \ge q_1 + 2 $. With 3 variables, minimum span is 3 (e.g., 1,3,5).

["Title: Understanding $q_3 = p_3 - 2 \le 8$, with Constraints: $q_1 \ge 1$, $q_2 \ge q_1 + 1$, $q_3 \ge q_2 + 1$, $q_3 \ge q_1 + 2$ — Finding All Valid Triplets", "Meta Description:\nExplore the integer solutions for $q_3 = p_3 - 2 \le 8$ under strict constraints: $q_1 \ge 1$, $q_2 \ge q_1 + 1$, $q_3 \ge q_2 + 1$, and $q_3 \ge q_1 + 2$. Discover how these rules define a minimum range of 3 across triplet values (e.g., 1, 3, 5).", "---", "### Introduction", "Math problems involving sequences with strict inequalities can seem complex at first, but by carefully analyzing constraints, we uncover elegant patterns and valid solutions. In this article, we drill into the inequality:", "$$\nq_3 = p_3 - 2 \le 8\n$$\nsubject to the rules:\n- $ q_1 \ge 1 $\n- $ q_2 \ge q_1 + 1 $\n- $ q_3 \ge q_2 + 1 $\n- $ q_3 \ge q_1 + 2 $", "Though $ q_3 $ is defined as $ p_3 - 2 $, we simplify by letting $ q_3 = p_3 - 2 $ and focus on the full system of inequalities. Despite the placeholder $ p_3 $, solving purely under the defined variables reveals a structured set of valid integer triplets $(q_1, q_2, q_3)$ with a minimum span of 3 — for example: $ (1, 3, 5) $.", "---", "### Step 1: Reformulate Constraints for Clarity", "We work directly with:\n- $ q_3 \le 8 $ (since $ q_3 = p_3 - 2 \le 8 $ → $ q_3 \le 8 $)\n- $ q_1 \ge 1 $\n- $ q_2 \ge q_1 + 1 $ → $ q_2 > q_1 $\n- $ q_3 \ge q_2 + 1 $ → $ q_3 > q_2 $\n- $ q_3 \ge q_1 + 2 $ → $ q_3 \ge q_1 + 2 $", "Our goal: Find all integer triplets $(q_1, q_2, q_3)$ satisfying all constraints, with minimum span meaning the difference $ q_3 - q_1 \ge 2 $, but due to $ q_3 > q_2 > q_1 +1 $, this naturally enforces a spread of at least 3 in the sequence.", "Example valid triplet: $ (1, 3, 5) $ —\nCheck:\n- $ q_1 = 1 \ge 1 $ ✔️\n- $ q_2 = 3 \ge 1 + 1 = 2 $ ✔️\n- $ q_3 = 5 \ge 3 + 1 = 4 $ ✔️\n- $ q_3 = 5 \le 8 $ ✔️\n- Differences: $ q_2 - q_1 = 2 $ (≥1), $ q_3 - q_2 = 2 $ (≥1), and $ q_3 - q_1 = 4 \ge 2 $ — satisfies all", "---", "### Step 2: Why Minimum Span of at Least 3?", "Even though some variables differ by 1 or 2, the combined chain of constraints creates a non-degenerate gap:", "- $ q_2 \ge q_1 + 1 $ → $ q_2 \ge 2 $ if $ q_1 = 1 $\n- $ q_3 \ge q_2 + 1 \ge 3 $\n- Also $ q_3 \ge q_1 + 2 $. For $ q_1 = 1 $, $ q_3 \ge 3 $, but both must align with $ q_3 \ge q_2 + 1 \ge 3 $, and if $ q_2 = 3 $, $ q_3 \ge 4 $. So $ q_3 = 4 $ or more — nearest integer $ \ge 4 $, but could of course be larger.", "But crucially:\nSince $ q_3 > q_2 > q_1 + 1 $, and $ q_1 \ge 1 $, then:\n- Smallest possible $ q_1 = 1 $\n- $ q_2 \ge 2 $, but $ \ge q_1 + 1 = 2 $ → so $ q_2 \ge 2 $\n- $ q_3 \ge q_2 + 1 \ge 3 $, and $ \ge q_1 + 2 = 3 $", "Now, can $ q_3 = 3 $? Only if $ q_2 = 2 $ (minimum), $ q_1 = 1 $:\n- $ q_3 = 3 $ → $ 3 \ge 2 + 1 = 3 $ ✔️\nBut $ q_3 = 3 $, $ q_1 = 1 $ → $ q_3 - q_1 = 2 $ — satisfies $ q_3 \ge q_1 + 2 $", "Wait — is $ (1, 2, 3) $ valid?\n- $ q_1 = 1 \ge 1 $ ✔️\n- $ q_2 = 2 \ge 1 + 1 = 2 $ ✔️\n- $ q_3 = 3 \ge 2 + 1 = 3 $ ✔️\n- $ q_3 = 3 \le 8 $ ✔️", "But differences: $ q_2 - q_1 = 1 $ (minimum allowed), $ q_3 - q_2 = 1 $ (also minimum allowed), but is the span 3 - 1 = 2, and the values span 3 indices: 1, 2, 3 — so the minimum span (difference between first and last) is 2, not 3?", "Wait — contradiction in definition?", "But recall:\nWe require $ q_3 \ge q_1 + 2 $. For $ q_1 = 1 $, $ q_3 \ge 3 $.\nBut $ q_3 $ can be equal to $ q_1 + 2 $, so $ q_3 = 3 $, $ q_1 = 1 $, difference 2.", "But the minimum span condition says “minimum span of 3” — this seems ambiguous.", "Clarify:\nThe phrase "with a minimum span of 3" in triplets $(q_1, q_2, q_3)$ usually means the range from smallest to largest is at least 3, i.e., $ q_3 - q_1 \ge 3 $. However, our constraints do not require $ q_3 - q_1 \ge 3 $, only $ q_3 \ge q_1 + 2 $ and $ q_3 \le 8 $.", "Example: $ (1, 2, 3) $ → $ q_3 - q_1 = 2 $, span = 2 — but it satisfies all constraints.", "But perhaps “minimum span of 3” in the problem implies strictly increasing values with gaps so the set covers at least 3 consecutive or spaced indices.", "Recheck original: “…with $ q_3 - q_1 \ge 2 $, $ q_2 \ge q_1 + 1 $, $ q_3 \ge q_2 + 1 $, and $ q_3 \ge q_1 + 2 $” — no explicit span constraint. But the sample $ (1,3,5) $ hints a minimal spread of 4 (from 1 to 5).", "Ah — the example given is 1, 3, 5, span = 4. But the minimum span of 3 likely refers to the minimum required separation via constraints, not hard bounds.", "Wait — reconsider: the actual constraint chain:\nFrom $ q_1 \ge 1 $, $ q_2 \ge q_1 + 1 $, $ q_3 \ge q_2 + 1 \ge q_1 + 2 $, and $ q_3 = p_3 - 2 \le 8 $. But $ q_3 $ → $ q_1 + 2 $, so $ q_3 - q_1 \ge 2 $ — but can be exactly 2.", "However, to enforce a minimum gap across the triplet, suppose we interpret “minimum span of 3” as $ q_3 - q_1 \ge 3 $ — otherwise valid triplets with span 2 exist.", "But $ (1,2,3) $: $ q_1=1, q_3=3 $, $ \ge 3 $? No, $ 1+2=3 $, so $ q_3 \ge 3 $ — equality allowed → valid.", "So span can be 2. But the example triplet $ (1,3,5) $ has span 4.", "Clarify purpose: likely the problem intends to emphasize non-trivial increasing sequences satisfying strict relative gaps, ensuring values don’t cluster.", "Therefore, we focus on valid triplets under all logical constraints, and confirm that some have span ≥3, others span 2, but the structure enforces meaningful separation.", "But the key insight lies in the relative order and differences, not fixed span.", "---", "### Step 3: Systematic Enumeration Under Constraints", "Let $ q_1 = a $, where $ a \ge 1 $ — integer.\nThen:\n- $ q_2 \ge a + 1 $ → let $ q_2 = a + d_1 $, $ d_1 \ge 1 $\n- $ q_3 \ge q_2 + 1 = a + d_1 + 1 $, and $ q_3 \ge a + 2 $ → so $ q_3 \ge \max(a+2, a + d_1 + 1) $", "But $ a + 2 $ vs $ a + d_1 + 1 $:\n- If $ d_1 = 1 $, $ a + d_1 + 1 = a + 2 $ → both equal → $ q_3 \ge a+2 $\n- If $ d_1 \ge 2 $, $ a + d_1 + 1 > a+2 $ → $ q_3 \ge a + d_1 + 1 $", "Also $ q_3 \le 8 $", "So for fixed $ a \ge 1 $, $ d_1 \ge 1 $, $ q_3 \in [ \max(a+2, a + d_1 + 1),\ 8 ] $", "We now generate valid triplets by iterating small $ a $:", "---", "Case $ a = 1 $:\n- $ q_2 \ge 2 $ → $ q_2 = 2,3,4,\dots $, $ \le 8 - \ ext{some } q_3 $\n- $ q_3 \ge q_2 + 1 $, $ q_3 \le 8 $, $ q_3 \ge 1 + 2 = 3 $", "But stricter: $ q_3 \ge q_2 + 1 $, $ q_2 \ge 2 $ → $ q_3 \ge 3 $", "So:\n- $ q_2 = 2 $ → $ q_3 \ge 3 $, $ q_3 \le 8 $ → $ q_3 = 3 $ to 8 → 6 values\n- $ q_2 = 3 $ → $ q_3 \ge 4 $, $ q_3 \le 8 $ → 5 values\n- $ q_2 = 4 $ → $ q_3 \ge 5 $ → 4\n- $ q_2 = 5 $ → $ q_3 \ge 6 $ → 3\n- $ q_2 = 6 $ → $ q_3 \ge 7 $ → 2\n- $ q_2 = 7 $ → $ q_3 \ge 8 $ → 1\n- $ q_2 = 8 $ → $ q_3 \ge 9 $, but $ q_3 \le 8 $ → invalid", "So for $ q_1 = 1 $:\n- $ q_2 $ from 2 to 7\n- For each: $ q_3 $ from $ q_2 + 1 $ to 8", "Total triplets: $ 6 + 5 + 4 + 3 + 2 + 1 = 21 $ triplets starting with $ q_1 = 1 $", "---", "Case $ a = 2 $:\n- $ q_1 = 2 $\n- $ q_2 \ge 3 $, $ q_3 \ge q_2 + 1 \ge 4 $, $ q_3 \ge 2 + 2 = 4 $, so $ q_3 \ge 4 $", "- $ q_2 = 3 $ → $ q_3 \ge 4 $, ≤8 → 5 values\n- $ q_2 = 4 $ → $ q_3 \ge 5 $ → 4\n- $ q_2 = 5 $ → $ q_3 \ge 6 $ → 3\n- $ q_2 = 6 $ → $ q_3 \ge 7 $ → 2\n- $ q_2 = 7 $ → $ q_3 \ge 8 $ → 1\n- $ q_2 = 8 $ → $ q_3 \ge 9 $ → invalid", "Total: $ 5 + 4 + 3 + 2 + 1 = 15 $ triplets", "---", "Case $ a = 3 $:\n- $ q_1 = 3 $\n- $ q_2 \ge 4 $, $ q_3 \ge q_2 + 1 \ge 5 $, $ q_3 \ge 3 + 2 = 5 $", "- $ q_2 = 4 $ → $ q_3 \ge 5 $ → 4\n- $ q_2 = 5 $ → $ q_3 \ge 6 $ → 3\n- $ q_2 = 6 $ → $ q_3 \ge 7 $ → 2\n- $"]