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Another rational cryptarithm (Posted on 2006-02-22) Difficulty: 3 of 5

Replace each variable with a positive rational number (i.e. fraction) such that all of the following equations simultaneously hold true. Note that 0/n is not considered positive.

O + N + E         = 1
T + W + O         = 2
T + H + R + E + E = 3
F + O + U + R     = 4
F + I + V + E     = 5
S + I + X         = 6
S + E + V + E + N = 7
E + I + G + H + T = 8

Additional constraints:

1) All fractions must have the same denominator, which must be as small as possible.
2) All variables must be distinct.
3) Reducible fractions, i.e. those in which the numerator and denominator share a common divisor greater than 1, are not allowed.

Hint: there are four distinct solutions. For extra credit, find the one with the lowest possible sum of the 14 variables.

This problem was inspired by a similar submission by pcbouhid (see

No Solution Yet Submitted by Ethan    
Rating: 2.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
A solution | Comment 1 of 5

Using a denominator of 83:

O = 12, N= 35, E=36, T=52, W=102, H=63, R=62, F=80, U=178, I=105, V=194, S=280, X=113, G=408.

It strikes me that more than four distinct solutions are possible. With four independant variables (G, U, W, X) solutions should be abundant.  I chose a relatively large random prime number as the denominator and found an answer. 

Looking for more, with a smaller denominator.

Added with edit:

Using a denominator of 79

O=11, N=33, E=35, T=49, W=98, H=60, R=58, F=77, U=170, I=101, V=182, S=268, X=105, G=387

Looking for the smallest denominator.

Edited on February 22, 2006, 5:28 pm
  Posted by Leming on 2006-02-22 17:19:37

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