Puzzle of the Week for 11 January 1999: Solution (details)

Two-bit strategy:

  1. Test bit 1 at 1100 rpm. If it fails, test bit 2 at 100, 200, 300, ..., etc., stopping if a test fails or after testing at 1000 rpm, whichever happens first. At most 11 tests are required in this case (if the top speed is 900 or 1000 rpm).
  2. Test bit 1 at 2100 rpm. If it fails, test bit 2 at 1200, 1300, 1400, ..., etc., stopping at or before 2000 rpm as in the previous step. Again, at most 11 tests are required (2 tests with bit 1 and up to 9 with bit 2).
  3. Test bit 1 at 3000 rpm. If it fails, test bit 2 at 2200, 2300, 2400, ..., etc.
  4. Test bit 1 at 3800 rpm. If it fails, test bit 2 at 3100, 3200, 3300, ..., etc.
  5. Test bit 1 at 4500 rpm. If it fails, test bit 2 at 3900, 4000, 4100, ..., etc.
  6. Test bit 1 at 5100 rpm. If it fails, test bit 2 at 4600, 4700, 4800, ..., etc.
  7. Test bit 1 at 5600 rpm. If it fails, test bit 2 at 5200, 5100, 5200, ..., etc.
  8. Test bit 1 at 6000 rpm. If it fails, test bit 2 at 5700, 5800, and 5900 rpm.

Three-bit strategy:

  1. Test bit 1 at 2200 rpm. If it fails:
    1. Test bit 2 at 600 rpm. If it fails, test bit 3 at 100, 200, 300, ..., etc. At most 7 tests are required (1 each with bits 1 and 2, and up to 5 with bit 3).
    2. Test bit 2 at 1100 rpm. If it fails, test bit 3 at 700, 800, 900, and 1000 rpm.
    3. Test bit 2 at 1500 rpm. If it fails, test bit 3 at 1200, 1300, and 1400 rpm.
    4. Test bit 2 at 1800 rpm. If it fails, test bit 3 at 1600 and 1700 rpm.
    5. Test bit 2 at 2000 rpm. If it fails, test bit 3 at 1900 rpm.
    6. Test bit 2 at 2100 rpm.
  2. Test bit 1 at 3800 rpm. If it fails:
    1. Test bit 2 at 2700 rpm. If it fails, test bit 3 at 2300, 2400, 2500, and 2600 rpm.
    2. Test bit 2 at 3100 rpm. If it fails, test bit 3 at 2800, 2900, and 3000 rpm.
    3. Test bit 2 at 3400 rpm. If it fails, test bit 3 at 3200 and 3300 rpm.
    4. Test bit 2 at 3600 rpm. If it fails, test bit 3 at 3500 rpm.
    5. Test bit 2 at 3700 rpm.
  3. Test bit 1 at 4900 rpm. If it fails:
    1. Test bit 2 at 4200 rpm. If it fails, test bit 3 at 3900, 4000, and 4100 rpm.
    2. Test bit 2 at 4500 rpm. If it fails, test bit 3 at 4300 and 4400 rpm.
    3. Test bit 2 at 4700 rpm. If it fails, test bit 3 at 4600 rpm.
    4. Test bit 2 at 4800 rpm.
  4. Test bit 1 at 5600 rpm. If it fails:
    1. Test bit 2 at 5200 rpm. If it fails, test bit 3 at 5000 and 5100 rpm.
    2. Test bit 2 at 5400 rpm. If it fails, test bit 3 at 5300 rpm.
    3. Test bit 2 at 5500 rpm.
  5. Test bit 1 at 6000 rpm. If it fails:
    1. Test bit 2 at 5800 rpm. If it fails, test bit 3 at 5700 rpm.
    2. Test bit 2 at 5900 rpm.

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