Seventy! In order to fulfill the task, the white king has to capture every black piece but the king while stepping carefully around the guarded squares.

Ka4 2. 2. Ka5 3. Ka6 4. Kxa7 5. Kb6 6. Kc7 7. Kd7 8. Ke7 9. Kf7 10. Kg6 11. Kf5 12. Kg4 13. Kg3 14. Kf2 15. Ke1 16. Kd1 17. Kxc1 18. Kd1 19. Ke1 20. Kf2 21. Kg3 22. Kg4 23. Kf5 24. Kg6 25. Kf7 26. Ke7 27. Kd7 28. Kc7 29. Kb6 30. Ka5 31. Ka4 32. Ka3 33. Ka2 34. Kxa1 35. Ka2 36. Ka3 37. Ka4 38. Ka5 39. Kb6 40. Kc7 41. Kd7 42. Ke7 43. Kf7 44. Kg7 45. Kxh8 46. Kg8 47. Kf7 48. Ke8 49. Kd8 50. Kc8 51. Kb8 52. Kxa8 53. Kb7 54. Kc6 55. Kd5 56. Ke5 57. Kf5 58. Kg4 59. Kg3 60. Kg2 61. Kxh1 62. Kh2 63. Kh3 64. Kg4 65. Kh5 66. Kh6 67. Kxh7 68. Kg6 69. Kxg5 70. Kf4

From Stephen Addison, The Book of Extraordinary Chess Problems, 1989.