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Programming Enigma Puzzles

5 October 2015

Posted by on **From New Scientist #2355, 10th August 2002** [link]

In the numbers ENIGMA, MEANING and IDEA digits have been consistently replaced by letters, with different letters used for different digits. I have tested the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 to see if they are a common factor of both ENIGMA and MEANING.

Of course 1 is a common factor of them, and it turns out that nine of the numbers 1-12 are common factors of ENIGMA and MEANING (and indeed the other three numbers are factors of IDEA).

What is the numerical value of ENIGMA?

[enigma1199]

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This brute force approach runs in 721ms.

Solution:ENIGMA = 597240.This value of ENIGMA is the only value for which ENIGMA and MEANING share 9 divisors between 1 and 12.

The shared divisors are: 1, 2, 3, 4, 6, 7, 8, 9, 12.

Fortunately choosing D = 1 gives IDEA = 7150. Which is divisible by the missing divisors: 5, 10, 11.

The three numbers not common factors of ENIGMA and MEANING are either

a) 5,7,10, so 70 doesn’t divide ENIGMA and MEANING but does divide IDEA

or

b) 5,10,11 so 110 doesn’t divide ENIGMA and MEANING but does divide IDEA

In either case A=0, so 10 divides ENIGMA

so a) 8*9*11*5 divides ENIGMA, 8*9*11 divides MEANING, 70 divides IDEA

or b) 8*9*7*5 divides ENIGMA, 8*9*7 divides MEANING, 110 divides IDEA