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Reknit Tinker(ed) Ratio (Posted on 2009-10-04) Difficulty: 3 of 5
In this alphametic equation, each of the capital letters in bold denotes a different base ten digit from 0 to 9, and x is any positive integer ≤ 16. None of R and T can be zero.

REKNIT        9*x
---------- = -----------
TINKER      9*x + 1

Determine the minimum value of REKNIT and maximum value of TINKER.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts Analytics interruptus (spoiler) Comment 4 of 4 |
1) Let b = REKNIT and a = TINKER. 

    Then b/a = 9x/(9x + 1)
            a/b = (9x + 1)/9x = 1 + (1/9x)
            a/b - 1 = 1/9x
            (a-b)/b = 1/9x
            b/(a-b) = 9x

Therefore b must be divisible by 9 and x.
But if b is divisible by 9, then SOD (sum of digits) is divisible by 9.
The SOD(b) = SOD(a), because they are palindromes, so a is divisible by 9.
But if a and b are divisible by 9, then so is (a-b).
Therefore, b must be divisible by 81x.

Let b = 81xy.
Then (a-b) = 9y

2)  so what is y?  Most values of y just won't do.  We need repeats (mod y) among 1, 10, 100, 1000, 10000 and 100000, so I suspected y would need to involve multiples of 9 or 99 or 999 or its' factors 11 or 111

3) 
     (a - b) = TINKER - REKNIT =
               = (T-R)*99999 + (I-E)*9990 + (N-K)*900
     
     so y =  (T-R)*11111 + (I-E)*1110 + (N-K)*100

4) And that's as far as I got analytically.    In fact, in all the solutions found, y = 11*111 = 1221 = -11111 +9*1110 - 100. This time, I think the computer approach trumps the analytic approach.  Unless somebody can improve on my work?

  Posted by Steve Herman on 2009-10-07 17:48:43
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