The answer to the latest international maths Kangaroo multiple choice challenge is 3435. This puzzle, part of a competition taken by more than seven million children annually, requires finding a four digit number that equals the sum of its digits raised to their own powers.
The 3435 Solution
The equation asks for a number abcd where the value equals a^a + b^b + c^c + d^d. In this notation, abcd represents the digits in their respective places, not a multiplication of a, b, c, and d. The digits do not need to be distinct.
The correct number is 3435. When you calculate 3^3 + 4^4 + 3^3 + 5^5, the result is exactly 3435. This specific combination of digits is the only one that satisfies the condition among the multiple choice options provided in the quiz.
How the Digits Are Determined
Deriving the answer requires establishing strict bounds for each digit. The solver first notes that 6^6 exceeds the maximum possible value for a four digit number. This means every digit in the solution must be less than 6.
One digit must be 5. If all digits were 4 or lower, the maximum sum would be 4^4 + 4^4 + 4^4 + 4^4, which equals 1024. This does not match the structure of a number starting with 1, 2, 3, or higher in the thousands place that would result from higher powers. Testing 5555 fails because it does not equal 4 times 3125.
The first digit, a, is determined to be 3. The logic eliminates 9 because the resulting sum would not align with the digit 9 in the thousands place. It also eliminates 6 because the sum would require digits that do not match the number 6250 plus combinations of smaller powers. This leaves 3 as the only viable option for the first digit, placing the number in the 3000 range.
With a fixed at 3, the remaining digits b, c, and d must include a 5. Testing combinations reveals that 3435 is the only fit. Other combinations like 3409 or 3412 are eliminated because they contain digits not allowed by the power constraints or lack the necessary 5. The final verification confirms that 3^3 + 4^4 + 3^3 + 5^5 equals 3435.
The International Maths Kangaroo is a major global event. The UKMT handles entries for the United Kingdom, while other countries have their own national associations. The problems are drawn from the World Olympiads Bank, an archive created by Sokratis Romanidis containing nearly 100,000 problems from international competitions in mathematics, physics, chemistry, biology, informatics, and astronomy.
Source: The Guardian
