BrainDen.com - Brain Teasers
• 0

# Champernowne Puzzle

## Question

Which of these is larger?

the total number of digits in the sequence 1, 2, 3, ..., 101180

or

the number of zero digits in the sequence 1, 2, 3, ... , 101181

## Recommended Posts

• 0

It's the same.

The number of digits in the first range is SUMn=[1,1180](9*10n-1*n) + 1181. 1180] into sub ranges based on the number of digits and call the number of digits n.

Let's break the range [1..10

For single digit numbers (n=1) the sub range is 1..9. There are 9 total digits.

For n=2, the sub range is 10-99. It contains 90*2 or 90*n digits.

For n=3, the sub range is 100-999. It contains 900*3 or 900*n digits

It's easy to see that for each sub range n the total number of digits is 9*n*10n-1

Adding up these sub ranges accounts for all except the last number, which has 1181 digits, so we need to add it to the sum.

Similarly, the number of zeroes in the second range is SUMn=[2,1181](9*10n-2*(n-1)) + 1181, so they are the same.

## Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account. ×   Pasted as rich text.   Paste as plain text instead

Only 75 emoji are allowed.

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.