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#75 Letters and numbers

Set by Christopher Dearlove

Let each letter have a value, where the vowels A, E, I, O and U have value zero and all other letter values are whole numbers greater than zero. The value of a word is the sum of the values of its letters. A word that represents a number is 鈥渟elf-describing鈥� if its value is that number. For example, ONE is self-describing if ONE = 1, for which we must have N = 1.

If TWO and THREE are also self-describing, which other letters can we find the values of?

If we continue with FOUR, etc., being self-describing and retain all deduced letter values, which is the first number word that cannot be self-describing?

What further letter values can we find, following the same rules, skipping over those number words that aren鈥檛 self-describing but keeping all earlier letter deductions, up to the word TWENTY?

Solution next week

#74 Triple digits

Solution

The smallest sum possible is 774, which can be made by, for example, 147 + 258 + 369 = 774. The second smallest is 783. One way to do this is 146 + 258 + 379 = 783. Swapping two digits between columns always changes the total by a multiple of 9. For example, to go from 774 to 783, we can swap the 6 in the tens column of 369 with the 7 in the units column of 147 (to get 146 and 379) 鈥� so the total rose by 10 and fell by 1, increasing it by 9 overall. Since the smallest total (774) is a multiple of 9 and moving digits changes the total by a multiple of 9, all possible totals are multiples of 9 鈥� so it is impossible to get to 1000. The closest possible total is 999, and can be made as, for instance, 289 + 576 + 134.