Hi Sarah,

I can help you get started. Sam had to stop for lunch after he had written $555$ digits. What is the last digit he write before lunch?

First you have to decide if zero is a natural number. Some authors include zero and some do not. You need to check with your textbook or your teacher. I am going to say zero is a natural number. Hence Sam starts

\[0 \: 1\: 2 \: 3 \: ... \: 9.\]

That's 10 one digit numbers for a total of 10 digits. He continues through the 2 digit numbers

\[10 \: 11 \: 12 \: ... \: 99.\]

So now he has written 100 numbers (0 to 99), 10 of which were 1 digit numbers so he wrote $100 - 10 = 90$ two digit number. Thus so far he has written

\[10 + 2 \times 90 = 190 \mbox { digits.}\]

Before lunch he writes $555$ digits so he has $555 - 190 = 365$ digits to go. The next numbers are 3 digit numbers and $365 \div 3 = 121$ with a remainder of $2$ and hence he writes $121$ three digit numbers and the two more digits. The three digit numbers he write are $100, 101, ..., 220$ and the next two digits he writes are $2 \:2$ and thus the last digit he writes before lunch is $2.$

After lunch he continued. What is the $6001^{st}$ digit Sam wrote?

Penny