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Maxim, Minerva & Kangaroo (Posted on 2017-04-22) Difficulty: 3 of 5
The twins Maxim & Minerva independently replaced the letters in the word KANGAROO upon the following rules:
1. Similar letters are replaced by the same digit
2. The resulting 8-digit number must be divisible by 11.
3. Maxim had to choose the maximal qualifying number, Minerva - the minimal.

Comparing their correct answers they've found out that a certain letter in the word KANGAROO was replaced by the same digit in their independent answers.

Specify WHAT letter and explain WHY.

No Solution Yet Submitted by Ady TZIDON    
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Solution Solution | Comment 2 of 4 |
I'll assume the basics: no leading zeros and each letter represents a distinct base 10 digit.

For KANGAROO to be a multiple of 11, the difference (K+N+A+O)-(A+G+R+O) must also be a multiple of 11.  This difference simplifies to K+N-(G+R) = 0 mod 11.

Let Minerva's first three digits be the absolute minimum with the assumptions previously mentioned.  Then KAN=102.  This implies 3-(G+R) = 0 mod 11 with G and R limited to {3,4,5,6,7,8,9}.  The smallest G can be in this equation, given the set of digits available, is G=5, which makes R=9.  Then Minerva has KANGAROO=10250933.

Similarily, Maxim's first three digits are KAN=987.  This implies 16-(G+R) = 0 mod 11 with G and R limited to {0,1,2,3,4,5,6}.  Then the largest G is 5 with R=0.  Then Maxim has KANGAROO=98758066.

Then comparing the numbers is simple:
The letter/digit they have in common is G=5.

  Posted by Brian Smith on 2017-04-22 12:59:24
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