# Math puzzle - Running Out of Digits

You initially have 100 of each digit from 0 to 9. This means you have 1000 digits in total. This count for each digit is shown in the table below.

Digit 0 1 2 3 4 5 6 7 8 9
# Remaining 100 100 100 100 100 100 100 100 100 100

Now start counting by ones, from 1. Each time you say a number you must remove the digits required to make the number from your stockpile of digits. For example, after you have counted from 1 to 13, the above table now looks like:

Digit 0 1 2 3 4 5 6 7 8 9
# Remaining 99 94 98 98 99 99 99 99 99 99

What is the largest number you can count to without running out of the digits needed to form the number?

• Hi. You should probably clarify whether you have to use all of the numbers from 1 to the limit, or if numbers can be skipped. Feb 13, 2020 at 23:55

You can count to:

162

Because:

It's quite obvious that the 1 will be the most used digit, since it's the lowest non-zero digit. Now, there are twelve 1 from 1-20. After that, it's an additional one 1 per ten, up til 99, making it 20. Now, we continue adding 1 every count, in addition to the extra ones in 101, 121, 131, 141, 151, 161, a and 110-119.

It's basically:

21 + 9 for 1...99, then we add +1 for numbers 100...199, +1 for every tenth, 101, 111, 121, 131, ..., as well as the extra 9 ones for 110-119.

Check the edit history (it took too much vertical space) for an exhaustive demonstration (credit to @Avi):

• Beat me to it by four minutes - good work! Feb 13, 2020 at 15:26
• Pipped me by ~ 10 seconds :x
– Avi
Feb 13, 2020 at 15:26

My calculations are a little rushed, but I think the answer is:

162

Working:

The dominant factor is going to be 1s - in particular we use one for every number from 100 to 199, so we can't possibly reach 200. We use at least one 1 for every other digit we count, since 1 always comes up first.
I homed in by trial and error, but the trick I used was calculating each place separately:
In the hundreds place, we use 63 copies of 1, for the numbers 100 to 162.
In the tens place, we use 20 - 10 to 19 and 110 to 119 (we've already counted the 100s places).
In the ones place, we use one in each ten numbers, for 17 total - (0)1, 11, 21, 31, ... 161

If I haven't made an arithmetic error (like the one I caught while typing!) that gives us our answer

• Pipped by four minutes! Feb 13, 2020 at 15:24

perl -E"@a=(100)x10;L:say++$i;$a[$_]--for($i=~/./g);goto L unless grep{\$_<=0}@a"