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Math Magic Archive

Problem of the Month
(May 2016)

This month we investigate various problems involving number names:

1. What is the smallest non-negative integer whose name contains exactly n of a given letter?

2. Consider the following list of spelled-out numbers and letters:


Notice that each is a true statement of how many of that letter have appeared up to and including in that line, and the numbers in the statements are non-decreasing. Given the last lines are n occurrences of the same number, what are the shortest lists that accomplish this? (In breaking ties, we want the shortest number of statements, and then the numerically smallest list of numbers, and then the earliest alphabetical list of letters.) (This problem is a variation of a problem of Eric Angelini.)

3. It is well-known that TWELVE + ONE = ELEVEN + TWO is a true mathematical statement as well as an anagram. Other mathematical anagrams are not true mathematically, such as SIXTY + FIFTEEN being 9 more than FIFTY + SIXTEEN. Given an integer n, what is the collection of number names that use the same letters, but differ in value by n, so that the values are as small as possible?

4. If we link the names for the first n numbers crossword style, what is the smallest rectangle (by area) they will fit into?

5. Is there a positive integer n with the following property: if digits are substituted for letters in the name of n (with spaces being ignored), and then the digits are interpreted in some integer base, the result is n. (I think the smallest number for which this evaluation can give a number smaller than n is the number ONE BILLION = 10234554106 = 10,816,350 < 1,000,000,000.) If no such number exists, then how good an approximation can we find (using relative error)?

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