A series of intriguing mathematical problems centered around the number 11 was recently presented, offering a blend of number theory and problem-solving challenges. These puzzles, originally devised by the University Maths Schools in the UK, explore divisibility rules, palindrome patterns, and the strategic arrangement of numbers. Here’s a breakdown of the puzzles and their solutions.
The Football Team Formation
The first problem involved dividing a football team’s players (numbered 1 to 11) into defenders, midfielders, and forwards such that the sum of shirt numbers within each group is divisible by 11. The solution? It’s impossible. The sum of numbers from 1 to 11 is 66, and excluding the goalkeeper (number 1) leaves a sum of 65 for the outfield players. Since the divisibility rule requires each group’s sum to be divisible by 11, and the total of these groups must also be divisible by 11, the impossibility arises because 65 is not divisible by 11.
Why this matters: Divisibility rules are fundamental in number theory and have practical applications in modular arithmetic and cryptography. This puzzle highlights how seemingly simple constraints can lead to mathematical impossibilities.
Palindromic Products of 11
The second problem explored palindromes formed by multiplying 11 by single-digit numbers (1 to 9). The challenge extended to finding additional palindromes when multiplying 11 by numbers up to 99. The solutions include four cases with matching digits (11, 22, 33, 44) and four “staircase” numbers (56, 67, 78, 89). Additionally, 11 x 91 = 1001 is also a palindrome.
Why this matters: The 11-times table’s unique palindromic property is a direct result of the base-10 number system. The multiplication process reveals patterns in digit sums and carries, making it a simple yet effective demonstration of mathematical structure.
The Largest Divisible Number
The final puzzle tasked participants with creating the largest possible 10-digit number using digits 0–9 once each, while ensuring divisibility by 11. The answer is 9876524130. The divisibility rule for 11 involves alternating addition and subtraction of digits. To verify, the odd positions (9, 7, 5, 4, 3) sum to 28, while the even positions (8, 6, 2, 1, 0) sum to 17. The difference (28-17 = 11) confirms divisibility.
Why this matters: This puzzle demonstrates how mathematical rules can be strategically applied to solve complex problems. The divisibility rule provides a quick and efficient method for verifying large numbers without performing full division.
The University Maths Schools, which developed these puzzles, are state sixth forms in the UK that cater to mathematically gifted students aged 16–19. For more information, visit umaths.ac.uk. These challenges serve as a testament to the elegance and accessibility of mathematical thinking.
