Turing machine deciding L = { 0ⁿ 1ⁿ : n ≥ 0 }

The machine repeatedly matches the leftmost unmarked 0 with the leftmost unmarked 1 to its right, marking them X and Y. States: q_scan (find the next 0), q_scan1 (in the 1-region: only matched 1s may remain), q_find1 (find the matching 1), q_return (walk back to the start). It accepts when all 0s are matched and only matched 1s / blanks remain; it rejects if a 0 has no matching 1 (more 0s), a 1 is left over (more 1s), or a 0 appears after a 1 (wrong order). This is a decidable but non-regular language, so the machine genuinely needs its tape as a counter.