## Assignment 19

Due Friday, April 12. (I know a few of you are planning on being gone this Friday. I’ll accept this on Monday from anyone without penalty.)

3. A combination lock on a safe requires you to know 5 numbers, each ranging from 0 to 63, and each thus requiring at most 6 digits to represent in binary. For example: $0_{10}=000000_{2}$ and $63_{10}=111111_{2}$. A spy is planning on sending you these numbers, each as a message $m$ that is 8 bits/digits in binary. The first two digits (on the left side of the binary number) form a parity check function $p(m)$ given by: $p(m)=\begin{cases}00\text{ if m has 1 or 0 ones}\cr 01\text{ if m has 2 or 3 ones}\cr 10\text{ if m has 4 or 5 ones}\cr 11\text{ if m has 6 ones}\end{cases}.$ For example, if you receive the message 11010111, then you know that the parity check function value is 11 while the message is 010111, but since these don’t match you know that the message has been scrambled and is not reliable. The correct parity function value for the message 010111 would be 10, meaning that the received message should be 10010111. This all being said, you receive the messages: $01010001,\quad 01110010,\quad 00000010,\quad 10100100,\quad\text{and}\quad 01010101.$ Which numbers, in base 10, are you sure are a part of the combination to the safe? Which are you not sure about?