Problem 17: Number Letter Counts

If the numbers \(1\) to \(5\) are written out in words: one, two, three, four, five, then there are \(3 + 3 + 5 + 4 + 4 = 19\) letters used in total.

If all the numbers from \(1\) to \(1000\) (one thousand) inclusive were written out in words, how many letters would be used?

NOTE: Do not count spaces or hyphens. For example, \(342\) (three hundred and forty-two) contains \(23\) letters and \(115\) (one hundred and fifteen) contains \(20\) letters. The use of "and" when writing out numbers is in compliance with British usage.

Solution

We need to note how many times each word appears. I use the term "century" to denote the ranges 1-99, 100-199, 200-299, and so on up to 900-999.

A trailing "one" pops up nine times every century: in the third, for example, it occurs in the numbers 201, 221, 231, 241, 251, 261, 271, 281, and 291. Likewise, the trailing "two", "three", and so on up to "nine" occurs nine times per century, for a total of 90 times each.
"Ten", "eleven", "twelve", and so forth up to "nineteen" all happen once a century, for a total of ten times each.
The tens-place words, "twenty", "thirty", "forty", and so forth up to "ninety" happen ten times per century, for a total of one hundred times each.
The word "and" doesn't appear in the first century, but appears in all but the first element of every successive century, for a total of \(9 \times 99 = 891\) occurrences.
The word "hundred" is present in every number from 100 to 999, for a total of 900 occurrences.
A leading "one" precedes every element of the second century, a leading "two" precedes every element of the third, and so on for the remaining single digit numbers.
Finally, we have a single occurrence of "one thousand".

Python Code

def p17():
	res = 0
	# trailing and leading single digits
	place1 = ['one', 'two', 'three', 'four', 'five', 'six', \
		'seven', 'eight', 'nine']
	res += sum(map(len, place1)) * 190
	# teen words
	place11 = ['ten', 'eleven', 'twelve', 'thirteen', \
		'fourteen', 'fifteen', 'sixteen', \
		'seventeen', 'eighteen', 'nineteen']
	res += sum(map(len, place11)) * 10
	# tens place words
	place10 = ['twenty', 'thirty', 'forty', 'fifty', 'sixty', \
		'seventy', 'eighty', 'ninety']
	res += sum(map(len, place10)) * 100
	# "and"
	res += len('and') * 891
	# hundred"
	res += len('hundred') * 900
	# and finally:
	res += len('one') + len('thousand')
	return res

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