Rook’s Halfway Signal
Rook’s signal
Rook is tuning an observatory dial toward a target reading of 8. The dial starts at . After each pulse, its new reading is halfway between its current reading and 8:
This is a recurrence: a rule that uses the current value to make the next value. The subscript tells the step number, so is the reading after 3 pulses.
Read a recurrence one step at a time
A subscript is a step label, not multiplication. In , the tells which reading you have. The label means the next reading.
Practice rule:
Rook’s rule works the same way: use one known reading to make exactly one next reading.
Page 2 · Decode the pattern
Keep your values exact; fractions are welcome.
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Use the recurrence to complete the table.
Reading Gap to the target 0 0 1 2 3 4 5 -
Describe what happens to the gap after each pulse. Then write a rule for the gap after pulses.
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Find without calculating one at a time. Explain how the gap pattern helps.