data/problems/219.yml
---
:id: 219
:name: Skew-cost coding
:url: https://projecteuler.net/problem=219
:content: "Let A and B be bit strings (sequences of 0's and 1's). \nIf A is equal
to the leftmost length(A) bits of B, then A is said to be a prefix of B. \nFor
example, 00110 is a prefix of 001101001, but not of 00111 or 100110.\n\nA prefix-free
code of size <var>n</var> is a collection of <var>n</var> distinct bit strings such
that no string is a prefix of any other. For example, this is a prefix-free code
of size 6:\n\n0000, 0001, 001, 01, 10, 11\n\nNow suppose that it costs one penny
to transmit a '0' bit, but four pence to transmit a '1'. \nThen the total cost
of the prefix-free code shown above is 35 pence, which happens to be the cheapest
possible for the skewed pricing scheme in question. \nIn short, we write Cost(6)
= 35.\n\nWhat is Cost(10<sup>9</sup>) ?\n\n"