I take the one- and two-digit binary numbers:
Now I shuffle them in some way and concatenate them together into one string of ones and zeros, like this:
Can you break up this string back into the original numbers? Turns out, this particular string cannot be broken up in a single way:
On the other hand, the string 011011 can only be broken up to make (0, 1, 10, 11).
Let's call strings like 011110 ambiguous and strings like 011011 unambiguous.
How many unambiguous strings are there? As there are only 24 ways to order the four numbers, we can quite quickly try all of them and see that there are seven unambiguous strings:
Of course, we're just starting. Let's define the function U(n, k) to be the number of unambiguous strings made up of base n numbers up to k digits. The example was in binary (n=2), and the numbers were up to two digits (k=2), so now we know:
What can you tell me about U? My original thought was about U(10, 2). How large is it?