Mathematical Puzzles: The Curious Case of the Number 11
Earlier today, we presented three fascinating mathematical problems that all revolve around the number 11. These puzzles challenge logical thinking and numerical skills, offering a delightful brain teaser for enthusiasts. Below, we revisit each problem with comprehensive solutions to unravel the mysteries.
1. The Football Team Formation Challenge
Imagine you are the coach of a football team where players wear shirt numbers from 1 to 11, with the goalkeeper assigned number 1. Your task is to divide the remaining ten outfield players into three groups: defenders, midfielders, and forwards. The goal is to arrange the team so that the sum of the shirt numbers in each group is divisible by 11. Is this possible?
Solution: It is not possible to achieve this arrangement. The total sum of all shirt numbers from 1 to 11 is 66. Since the goalkeeper wears number 1, the sum for the outfield players is 66 minus 1, which equals 65. If the sums for defenders, midfielders, and forwards were all divisible by 11, then their combined total would also be divisible by 11. However, 65 is not divisible by 11, proving that no such division exists.
2. Palindromic Products in the 11-Times Table
When learning times tables, the 11-times table is often noted for its simplicity, with products like 11, 22, and 99 being palindromes—numbers that read the same forwards and backwards. But what happens when we extend this up to 11 multiplied by 99? How many more palindromic answers can we find?
Solution: There are nine additional palindromic products beyond the basic ones. To understand this, consider how multiplying by 11 works for two-digit numbers. When no carrying is required, the product is formed by the first digit, the sum of the digits, and the second digit. For example, 11 × 52 equals 572, with the middle digit as 5+2=7.
- Matching Digits: When the two digits are identical, such as in 11, 22, 33, or 44, the products are 121, 242, 363, and 484—all palindromes. This works only when the middle digit remains below 10, limiting it to these four cases.
- Staircase Numbers: For numbers where the second digit is one more than the first, like 56, 67, 78, and 89, the products are 616, 737, 858, and 979, respectively, all palindromes.
- The Final Case: At higher values, the product becomes a four-digit number. The only hope for a palindrome is when the last digit is 1, as seen with 91, where 11 × 91 equals 1001, a palindrome.
In total, this yields nine more palindromic products, adding to the initial nine from the basic table.
3. Constructing the Largest 10-Digit Number Divisible by 11
There is a lesser-known divisibility rule for 11: take the digits of a number and alternately add and subtract them, starting with a plus. If the result is a multiple of 11, including zero, the number is divisible by 11. For instance, with 132, we calculate +1-3+2=0, confirming divisibility. Using each digit from 0 to 9 exactly once, what is the largest possible 10-digit number divisible by 11?
Solution: The largest such number is 9876524130. To derive this, start with the largest 10-digit number using digits 0-9 once: 9876543210. Apply the divisibility test: sum digits in odd positions (9,7,5,3,1) equals 25, and in even positions (8,6,4,2,0) equals 20, with a difference of 5, not a multiple of 11. Since the total sum of digits 0-9 is 45, the difference between odd and even sums cannot be zero; the nearest multiple of 11 is 11.
By preserving the descending prefix as much as possible, we find that starting with 98765 works. The digits 9,7,5 in odd positions and 8,6 in even positions give a difference of 7. The remaining digits 4,3,2,1,0 must contribute exactly 4. To maximise the number, place 4 and 3 in odd positions and 2,1,0 in even positions, yielding a difference of 4. Arranging them as 9876524130 results in odd-position sum of 28 and even-position sum of 17, with a difference of 11, confirming divisibility by 11.
These puzzles offer a fun exploration of mathematical concepts, showcasing the versatility of the number 11. They were provided by the University Maths Schools, a network of eleven state sixth forms in the UK for students aged 16-19 passionate about mathematics. For more information, visit their website. We hope you enjoyed this mental workout and look forward to more puzzles in the future.
