Puzzle of the Week for 11 January 1999: Solution (details)
Two-bit strategy:
- 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).
- 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).
- Test bit 1 at 3000 rpm. If it fails, test bit 2 at 2200, 2300,
2400, ..., etc.
- Test bit 1 at 3800 rpm. If it fails, test bit 2 at 3100, 3200,
3300, ..., etc.
- Test bit 1 at 4500 rpm. If it fails, test bit 2 at 3900, 4000,
4100, ..., etc.
- Test bit 1 at 5100 rpm. If it fails, test bit 2 at 4600, 4700,
4800, ..., etc.
- Test bit 1 at 5600 rpm. If it fails, test bit 2 at 5200, 5100,
5200, ..., etc.
- Test bit 1 at 6000 rpm. If it fails, test bit 2 at 5700, 5800,
and 5900 rpm.
Three-bit strategy:
- Test bit 1 at 2200 rpm. If it fails:
- 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).
- Test bit 2 at 1100 rpm. If it fails, test bit 3 at 700, 800,
900, and 1000 rpm.
- Test bit 2 at 1500 rpm. If it fails, test bit 3 at 1200, 1300,
and 1400 rpm.
- Test bit 2 at 1800 rpm. If it fails, test bit 3 at 1600 and 1700 rpm.
- Test bit 2 at 2000 rpm. If it fails, test bit 3 at 1900 rpm.
- Test bit 2 at 2100 rpm.
- Test bit 1 at 3800 rpm. If it fails:
- Test bit 2 at 2700 rpm. If it fails, test bit 3 at 2300, 2400,
2500, and 2600 rpm.
- Test bit 2 at 3100 rpm. If it fails, test bit 3 at 2800, 2900,
and 3000 rpm.
- Test bit 2 at 3400 rpm. If it fails, test bit 3 at 3200 and
3300 rpm.
- Test bit 2 at 3600 rpm. If it fails, test bit 3 at 3500 rpm.
- Test bit 2 at 3700 rpm.
- Test bit 1 at 4900 rpm. If it fails:
- Test bit 2 at 4200 rpm. If it fails, test bit 3 at 3900, 4000,
and 4100 rpm.
- Test bit 2 at 4500 rpm. If it fails, test bit 3 at 4300 and
4400 rpm.
- Test bit 2 at 4700 rpm. If it fails, test bit 3 at 4600 rpm.
- Test bit 2 at 4800 rpm.
- Test bit 1 at 5600 rpm. If it fails:
- Test bit 2 at 5200 rpm. If it fails, test bit 3 at 5000 and
5100 rpm.
- Test bit 2 at 5400 rpm. If it fails, test bit 3 at 5300 rpm.
- Test bit 2 at 5500 rpm.
- Test bit 1 at 6000 rpm. If it fails:
- Test bit 2 at 5800 rpm. If it fails, test bit 3 at 5700 rpm.
- Test bit 2 at 5900 rpm.
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