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Unique Number of Letters

Out of all whole numbers between one and five thousand, there is only one number that has a unique number of letters when you spell it out. What number is it?

Include spaces and dashes. For example, “twenty-four” and “two hundred” both have 11 letters.

Solution

Three thousand

Here is the reasoning:

• First, we rule out the relatively uniquely-named numbers between ten and twenty.
• “Eleven” has 6 letters, but so does “twelve”
• “Thirteen” has 8 letters, but so does “fourteen”, “eighteen”, and “nineteen”
• “Fifteen” has 7 letters, but so does “sixteen”
• “Seventeen” has 9 letters, but so does “forty-two”
• Next, recognize that all of the nonzero digits share a number of characters with other digits. So the number must end in zero.
• “One”, “two”, and “six” have 3 letters
• “Four”, “five”, and “nine” have 4 letters
• “Three”, “seven”, and “eight” have 5 letters
• Similarly, all of the tens share a number of characters with other numbers. So the number must end in two zeroes.
• “Ten” has 3 letters, but so does “one”
• “Forty”, “fifty”, and “sixty” have 5 letters
• “Twenty”, “thirty”, “eighty”, and “ninety” have 6 letters
• “Seventy” has 7 letters, but so does “fifteen”
• Similarly, all of the hundreds share a number of characters with other numbers. So the number must end in three zeroes.
• This leaves the thousands. Since we limited to numbers up to five thousand, that means “three thousand” has a unique number of letters (“seven thousand” and “eight thousand” are out of the range).
• Finally, confirm “three thousand” is the only number between 1-5000 with 14 letters.
• “Seventy-seven” is the longest 2-digit number, and it has only 13 letters.
• Out of the 3-digit numbers, the multiples of 100 are too short (tied for longest is “three hundred” with 13 letters) and all others are too long (tied for shortest is “two hundred ten” with 15 letters).

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