Earlier today, we presented three puzzles centered around the number 11. Here are the solutions, as promised.
The first puzzle involved a football team with shirt numbers 1 to 11, where the goalkeeper wears 1. The task was to divide the outfield players into defenders, midfielders, and forwards such that the sum of shirt numbers in each group is divisible by 11. However, the total sum of outfield numbers is 65, which is not divisible by 11, making such a division impossible.
The second puzzle explored palindromic results in the 11 times table up to 11 × 99. When multiplying a two-digit number with identical digits (11, 22, 33, 44) by 11, the products (121, 242, 363, 484) are palindromes. Additionally, numbers where the second digit is one greater than the first also yield palindromes. For four-digit products, 11 × 91 = 1001 is a palindrome.
The third puzzle asked for the largest 10-digit number using digits 0–9 exactly once that is divisible by 11. Using the divisibility test (alternating sum of digits must be a multiple of 11), the largest such number is 9876524130. This was achieved by arranging digits to ensure the alternating sum equals 11.
These puzzles were provided by the University Maths Schools, of which there are eleven across the UK. For more information, visit umaths.ac.uk.
