"No it isn't possible. The sum of numbers from 1 to 11 is 66. So the shirt number total for the outfield players is 66- 1 = 65. If the sums of shirt numbers of defenders, midfielders and forwards are all divisible by 11, the so is the sum of shirt numbers of all these three groups together. But we know this is false, since 11 does not divide 65."
"Consider how multiplying by 11 works. When a two-digit number has digits a and b, the product with 11 is formed by writing down the first digit, then their sum, then the second digit provided no carrying is needed. i.e. 11 52 = 572, since the middle digit is simply 5+2=7. i) matching digits (four solutions) If the two digits are the same, as in 11, 22, 33 or 44, then multiplying by 11 produces 121, 242, 363 and 484 all palindromes. This works only while the middle digit stays below 10, which limits us to these four cases."
The numbers 1 through 11 sum to 66, so the outfield players numbered 2–11 sum to 65, which is not divisible by 11; therefore the outfield players cannot be divided into three groups whose sums are each divisible by 11. For multiplication by 11, a two-digit number with digits a and b produces a three-digit number formed by a, (a+b), b when no carry occurs. That rule yields palindromes in several cases: four where the digits match (11, 22, 33, 44 producing 121, 242, 363, 484) and four staircase cases where the second digit is one greater than the first (56, 67, 78, 89 producing 616, 737, 858, 979). Counting these and one additional case gives nine further palindromes between 11×10 and 11×99.
Read at www.theguardian.com
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