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# Rec.Puzzles Hall of Fame

The rec.puzzles Hall of Fame is a compilation of over 500 of the most popular puzzles that have been posted and discussed in the rec.puzzles newsgroup. In most cases a detailed solution has been provided. Many of these puzzles also appear in Braingle's own collection.

Categories

## pickover : pickover.03.p

Title: Cliff Puzzle 3: Too many 3's From: cliff@watson.ibm.com If you respond to this puzzle, if possible please include your name, address, affiliation, e-mail address. If you like, tell me a little bit about yourself. You might also directly mail me a copy of your response in addition to any responding you do in the newsgroup. I will assume it is OK to describe your answer in any article or publication I may write in the future, with attribution to you, unless you state otherwise. Thanks, Cliff Pickover * * * How many numbers have at least one digit -- a three? In the first 10 numbers, 1,2,3,4,5,6,7,8,9,10 there is only one number which contain the digit 3. This means that 1/10 or 10% of the numbers have the number 1 in the first 10 numbers. In the first 100 numbers the occurrence of numbers with at least one three seems to be growing. In fact there are 19 numbers: 3,13,23,33,43,53,63,73,83,93, 30,31,32,34,35,36,37,38,39. This means that about 19% of the digits contain the number 3 in the first 100 numbers. We can make a table showing the percentage of numbers with at least one 3-digit for the first N numbers. N % 10 1 100 19 1000 27 10000 34 The percentages rapidly increase to 100% indicating that almost all of the numbers have a 3 in them! In fact, a formula describing the proportion of 3's can be written: 1-(9/10)**N. The proportion gets very close to 1 as N increases. Stop And Think 1. How can it be that almost all of the numbers have a 3 in them?

Solution

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