To solve this, we can use a combinatorial approach. We want to form an 8-digit number using the digits 3 and 4, with exactly three consecutive 3's. Let's break it down:

["Solving the 8-Digit Puzzle: Using a Combinatorial Approach to Count 8-Digit Numbers with Exactly Three Consecutive 3’s", "Every puzzle beginner and math enthusiast loves a well-defined challenge — and forming an 8-digit number using only the digits 3 and 4, with exactly three consecutive 3’s, is one such brain-teasing task. Whether you're optimizing for computational efficiency or mathematical precision, applying a combinatorial approach lets us break down the problem logically and accurately. In this article, we break down how to count valid 8-digit numbers composed solely of the digits 3 and 4 that contain exactly three consecutive 3’s — no more, no less.", "---", "## Why Use a Combinatorial Approach?", "Combinatorics is the mathematical study of counting and arranging discrete elements under specific constraints. Instead of trying every possible 8-digit combination of 3’s and 4’s — which would involve $2^8 = 256$ possibilities (feasible but unstructured) — we leverage patterns and logical groupings.", "For problems like counting valid digit sequences with recurring sequences (e.g., exactly three consecutive 3’s), combinatorial reasoning allows clear case analysis, pattern recognition, and careful exclusion of invalid configurations, resulting in accurate, efficient solutions.", "---", "## Problem Restatement", "- Create an 8-digit number.\n- Only digits allowed: 3 and 4.\n- The number must contain exactly three consecutive 3’s (e.g., 333), but no longer runs of 3’s (so 3333, 33333, etc., are invalid).\n- Digits at the ends (beginning or end) count equally — we are not restricted to internal runs.", "---", "## Step 1: Understand Placements of the Exact Subsequence "333"", "The key challenge is ensuring the run of three 3’s appears exactly once, in a contiguous block, and no longer runs or overlapping repeats occur.", "We’ll consider all valid positions where the substring "333" can occupy, accounting for length and overlap:", "- The sequence “333” requires 3 adjacent digits, so it can begin at position 1 through 6 (positions 1–6, since 6+3−1=8).\n- However, placing three 3’s must not extend into a longer run or create a second run.", "Let’s consider each valid starting index $i$ from 1 to 6.", "For each starting position $i$, construct a template with “333” fixed and analyze valid digit assignments for remaining positions, ensuring no extension of runs and no second “333” block.", "We define three non-overlapping cases based on run boundaries and digit constraints.", "---", "## Step 2: Classify Valid Patterns by Placement of "333"", "We categorize valid placements by:\n1. The starting index of “333”\n2. Whether it’s bounded on both sides by 4s (or edges) to prevent extension\n3. Avoiding double counting of overlapping sequences", "### Case 1: Isolated “333” (no adjacent 3s before or after)", "To ensure “333” appears only once, place "333" surrounded by 4s (or boundaries), e.g., 4 333 4.", "Valid templates:", "- Starts at position 1: 333**** → but first three are “333”; need positions 4–8 to avoid extending or starting new runs.\n → To prevent extension and other runs, positions 4,5,6 must not be 3, or else overlapping runs may form.\n → To guarantee exactly three 3s in a row and no longer runs, we require:\n - Position 4 = 4\n - Position 5 = 4\n - Position 6 = 4\n → Then positions 7 and 8 can be either 3 or 4 — but if we place a 3 in 7 or 8, must not form a new run of 3 or extend.", "Wait — but if we allow a 3 after the “333”, we risk starting a new run adjacent or overlapping.", "Better approach: isolate the “333” block with 4s on both sides, and restrict surrounding digits.", "#### Subcase: “333” at start with isolation", "Let’s fix:", "- Positions 1–3: 333\n- To stop extension to the left: no 3 at position 0 (not applicable)\n- To stop extension to right: position 4 must be 4 (already required)\n- To prevent forming another run: positions 4 and 5 must be 4 to avoid overlapping\n- Then positions 7 and 8: can place digits, ensuring no additional “333”", "#### Pattern: 3334** → positions 1–3: 333, pos 4 = 4", "Now, pos 5 = 4 (to prevent pos 3–5 = 333), pos 6,7,8 free? But wait — could pos 5–7 = 444? No run, but could pos 6–8 be 333? No — pos 5 is 4.\nBut now pos 6–8 can be 3? Let’s see:", "We must avoid any additional "333" substring.", "So, possible trailing digits: positions 5–8 must not contain “333” and must not extend run.", "So, fix: pos 4 = 4, pos 5 = 4\nThen pos 6: can be 3 or 4\n- If pos 6 = 3 → pos 7 and 8 cannot both be 3 (would make 3343 → no run of 3)\n → But pos 7–8 = 33 → allowed only if not completing three\n → So pos 6 = 3 is allowed, but pos 7 and 8:\n - If 3,3 → pos 6–8 = 333 → entire block → 4 digits → invalid (run longer than 3)\n - So pos 7 and 8 must not form 333 → so cannot be 3,3\n → So if pos 6 = 3, then pos 7 and 8 must both be 4\n- If pos 6 = 4 → pos 7 and 8 can be 3 or 4, as long as no 333 occurs — which is safe unless pos 7=pos8=pos9, but pos 9 doesn’t exist. So no issue.", "So breakdown:", "Pattern: 333444 → pos 6–8 = 444 → valid\nPattern: 3334443 → pos 6=3, 7=4, 8=3 → no run → valid? But check: pos 6=3, pos 7=4, pos 8=3 → no three in a row → valid\nBut wait: pos 7–8 = 43 — safe\nBut is 333–443 → runs are 333, then 443 → not extended → valid", "But can pos 7 = 3?\nOnly if not forming 333. But pos 6–8 = 333 → full run → invalid unless pos 6 ≠ 3", "So fix: pos 4 = 4 (left boundary), pos 5 = 4\npos 6: depends", "Case 1A: pos 6 = 4 → pos 7,8 free: 2^2 = 4 options → 443, 434, 443, 344 → all valid\n→ Total: 4 patterns\nCase 1B: pos 6 = 3\nThen pos 7: cannot be 3 (else 3343 → pos 6–8 = 333) → so pos 7 = 4\nThen pos 8 = 3 or 4 → allowed, no run\n→ One pattern: 3334334\n→ But check: 3334334 → contains “333” at start, then 334 → no longer run → valid\nSo 1 pattern for pos 6=3\n→ Total for Case 1A + 1B = 4 + 1 = 5 patterns", "But wait — pos 4 = 4, pos 5 = 4 fixed.", "So total for isolated “333” at beginning: 5 valid templates", "But wait — does pos 4 = 4 ensure no run extends? Yes — since pos 3 = 3, pos 4 = 4 → break.\npos 5 = 4 → ensures no run from before.\npos 6 ≥ 4 → no run from end unless three at end — but pos 7,8 can’t both be 3 if pos 6 ≠ 3.", "So yes, 5 valid sequences from this pattern.", "But we must ensure exactly one run of three — not multiple, not none else.", "We enforce isolation via 4s → so only one “333”", "Case 2: Central “333” (positions 2–4)\n → Pattern: 333**\nPositions:\n- 1: free (3 or 4)\n- 2–4: 333\n- 5: free\n- 6–8: free", "But to prevent extending runs:", "- Pos 1 ≠ 3 → else pos 1–3 = 333 → full run → invalid unless only 3 digits, but then may form more\n→ To avoid extension at start or end, enforce neighbor 4s where possible", "But pos 1 adjacent to pos 2 = 3 → if pos 1 = 3 → then pos 1–3 = 333 → multiple runs → invalid\n→ So pos 1 = 4", "Similarly, pos 5: adjacent to pos 4 = 3 → if pos 5 = 3 → then pos 2–4–5 = 3333 → run of 4 → invalid\n→ So pos 5 = 4", "Now remaining: pos 6,7,8 — can be 3 or 4, but must not create another “333”\n→ Possible substrings in pos 6–8: must not contain “333”\n→ Full length 3 → only one possibility:", "- 444 → valid\n- 443 → ok\n- 434 → ok\n- 433 → ok (only one 3 at end)\n- 333 → invalid (run >3)\n- 334 → pos 6=3,7=3,8=4 → no three → ok, but wait — pos 6–8 = 334 → no run → valid?\nBut pos 5 = 4, pos 6 = 3 → safe\nBut pos 6–8 = 334 → runs of 1 → ok\nBut wait — is this part of a longer run? No — only one 3 at pos 6, followed by 4.\nBut — pos 7 = 3, pos 8 = 4 → so 334 → no three → valid?\nBut — pos 6 = 3, but not adjacent to another 3? pos 5 = 4 → so only one 3 → ok", "Wait — but if pos 6 = 3, pos 7 = 3, pos 8 = 4 → then 334 → no run of 3 → valid", "But earlier we required pos 5 = 4 — done\npos 6: can be 3? Let's recompute validity", "We fixed:\n- pos 1 = 4 (to avoid extending left)\n- pos 2–4 = 333\n- pos 5 = 4 (to avoid extending right)\nNow pos 6,7,8: each 3 or 4, but no “333” in pos 6–8", "Number of valid 3-digit strings without “333”: total $2^3 = 8$, minus one (333) → 7", "But we must exclude any that contain “333” — only one substring possible: pos 6–8", "So invalid if pos 6=3, pos 7=3, pos 8=3\nAll others are safe", "So valid combinations: 8 − 1 = 7", "But wait — is that correct?\nExample: 334 → valid\n333 → invalid → excluded\n334 → allowed\nBut what about 343? pos 6=3,7=4,8=3 → ok\n434 → ok\n433 → ok\n344 → ok\n433 → pos 6=3,7=3,8=3 → 333 → invalid\n333 → invalid\n334 → ok\nSo total 7 valid for pos 6–8", "But pos 6,7,8 → 7 valid combinations", "Thus total for Case 2: pos 1=4, pos 5=4 → fixed\npos 6–8: 7 options → 7 patterns", "But — pos 6 may be 3, but not forming run — yes, as long as not adjacent 3s — but pos 5 = 4 → safe", "So 7 valid numbers in this central case", "Case 3: “333” at position 3–5 → *333\n→ pos 2 and 6 must not be 3 to prevent extension or forming another run", "So:\n- pos 2 ≠ 3 → pos 2 = 4\n- pos 6 ≠ 3 → pos 6 = 4", "Now free positions: pos 1,7,8", "Fix pos 2 = 4, pos 6 = 4", "Now pos 1: 3 or 4 → but check: if pos 1 = 3 → then pos 1–2–3 = 333 → multiple runs → invalid\n→ So pos 1 = 4", "Now pos 7,8: any combination, but no “333” possible in pos 7–9 (none) — only possible run: 7,8,9 — pos 9 none → max 2 → safe", "So pos 7 and 8: 4 options each → $2^2 = 4$", "Thus patterns: 444 333 44 → fixed → 44433344, 44433344 (pos7=4,8=4), pos7=4,8=3, pos7=3,8=4, pos7=3,8=3 → all valid?", "Wait — pos 7 = 3, pos 8 = 3 → ok, only two → no run of 3 → valid\nBut do they form run with anything? pos 6 = 3 → pos 6–7–8 = 333 → run of 3 → but pos 6 is inside? → 333 within positions 6–8 → yes!", "Wait — pos 6 = 3 (by constraint), pos 7 = 3 → so pos 6–7 = 333 → run of 2 → but pos 7–8 = 3,4? pos8 = 4 → so pos7–8 = 34 → safe → but pos6–7–8 = 333 → three in a row → invalid → run of three!", "Ah! Critical error.", "Because pos 6 = 3 (by isolation of central block), pos 7 = 3 → then pos 6–7–8 = 333 → run of three → invalid", "Similarly, pos7 = 3, pos8 = 3 → if pos6 = 3 → three run → invalid", "So to prevent this, since pos6 = 3, we must have pos7 = 4 to break", "Similarly, pos8 can be 3 or 4 — but if pos8 = 3, pos7 = 4 → safe (4–3)", "So revise:", "- pos 6 = 3 (fixed by isolation)\n- To avoid 333 at 6–8 → pos7 must be 4\n- pos8: can be 3 or 4 → both safe (no three possible)", "So pos 8: 2 options\npos 7: fixed to 4\npos 1: fixed to 4\npos 2: fixed to 4 (to avoid extension from left)", "So total: pos8 free (2 choices), others fixed → 2 patterns", "Examples:\n- 44433344 → pos6–8 = 333 → but pos7=4 → no run → wait: pos6=3,pos7=4,pos8=4 → so 333 → only pos6–7? No — pos6–7–8 = 3,4,4 → no three → ok\nBut pos6–7–8 = 334? No — pos7=4 → so 334 → no run of 3 → valid\nBut pos6–7–8: 3,4,4 → ok — no three\nBut pos6=3, pos7=4 → so break", "But earlier: pos6–8 = 33? No — pos7=4 → so only two → so no run → valid", "Wait — but if pos6 = 3, pos7 = 4 → so 334 at start of block → no three → safe", "But pos6 = 3 → pos7 must be 4 → enforced\npos8: can be 3 → 3343 → ok", "So pos8: 2 options", "Thus total: 2 patterns for Case 3", "Summary of Combinatorial Cases:", "| Case | Position | Constraints | Free Variables | Count |\n|------|---------|------------|----------------|-------|\n| 1 | Starts at 1 | pos4=4, pos5=4; pos6–8: avoid “333” — 7 valid | pos6,pos7,pos8 | 7 |\n| 2 | Pos 2–4 | pos1=4, pos5=4; pos6–8"]