Rook’s Halfway Signal

At a glance

The readings are 0,4,6,7,152,314,…. The key change of viewpoint is to track the gap 𝑔𝑛 =8 −𝑥𝑛. Every pulse halves that gap, so 𝑔𝑛 =82𝑛 and 𝑥𝑛 =8 −82𝑛. Therefore 𝑥10 =1023128 =7127128.

Key idea

The recurrence looks as if each new reading depends on all the work before it. Measuring the distance from the fixed target turns it into a familiar geometric pattern:

Thus the gap, not the reading, is the simplest quantity to follow.

Worked solution, part by part

1. Complete the table

Each reading is obtained by substituting the previous reading into the supplied rule.

2. Describe and model the gap

The gap is divided by 2 after every pulse. Since the first gap is 𝑔0 =8,

Equivalent descriptions such as 𝑔𝑛+1 =12𝑔𝑛, together with the starting value 𝑔0 =8, are also correct.

3. Find the tenth reading

At step 10, the gap is

Subtract this from the target:

Check

Every listed reading is below 8, while each gap is positive and half the preceding gap. The formula gives 𝑥0 =8 −8 =0, so it also matches the starting value.