324 // Professor Stewart’s Superlative Sneaky Solutions
The Millionth Digit The millionth digit is 1. The numbers 1–9 occupy the first 9 positions. The numbers 10–99 occupy the next 2690 ¼ 180 positions. The numbers 100–999 occupy the next 36900 ¼ 2,700 positions. The numbers 1,000–9,999 occupy the next 469,000 ¼ 36,000 positions. The numbers 10,000–99,999 occupy the next 5690,000 ¼ 450,000 positions. At this stage, we have reached the 488,889th digit altogether. Since 1,000,000 488,889 ¼ 511,111, we want the digit in the 511,111th place, in the block that starts 100,000-100,001-100,002- and so on. Since these are grouped in sixes, we work out 511,111/6 ¼ 85,185 16. Therefore we are seeking the first digit of the 85,186th 6-digit block.
........................................... That block must be 185,185, and its first digit is 1.
Piratical Pathways Redbeard’s bank is at 19 Taxhaven Street.
Calculating the number of paths.
The number is small enough that you can just list the possible paths, but there’s a systematic method for this sort of question. The diagram shows the same map; I’ve removed superfluous connections that can never be used to keep thing simple, but it makes no difference to the method or the result if you leave them in.