Mathematicians and Discoveries!

15 Points Per Correct Answer - No time limit  

1. Blaise Pascal, a French mathematician was credited for his works on Pascal's triangles and binomial coefficients. Pascal's triangle is a geometric representation of numbers in a triangular shape that displays the values of binomial coefficients when (x+y)^n is expanded, provided that n is a positive integer. Each number in the triangle is the sum of the two numbers that lie above it. The first line consists of a sole number 1. The second line has two 1's. From the third line onwards, the addition operations are required to produce the numbers 1, 2, 1. Meanwhile, the fourth line's numbers are 1, 3, 3, 1. What is the value of the number at the middle of the fifth line? More hint: this number is the first perfect number. Answer: (1-digit number)

2. This number's irrationality property was first discovered by Pythagoras and it was named after him. What is the value of Pythagoras' constant? The square root of ___. More hints: The only even prime number. Answer: (1-digit number)

3. In mathematics, Mersenne primes are prime numbers that are in the form of 2^n - 1, where n itself is another prime number. The first Mersenne prime is 2^2 - 1 = 3 and the second being 2^3 - 1 = 7. Pietro Cataldi, an Italian mathematician discovered the sixth and seventh Mersenne primes. The value of n of the sixth Mersenne prime is 17 and the prime number is 2^17 - 1 = 131071. What is the value of n of the seventh Mersenne prime? More clues: this number is smaller than 20. Answer: (2-digit number)

4. The Greek mathematician, Euclid, or also known as the Father of Geometry discovered the first four perfect numbers, the first three being 6, 28 and 496. What is the fourth perfect number? More clues: the first two digits is the largest two-digit square and the last two digit is the only two-digit perfect number. Answer: (4-digit number)

5. Perhaps this was not a discovery, but this eight-year old kid was indeed a child prodigy. Carl Friedrich Gauss, better known as the Prince of Mathematics, was asked to calculate the sum of all the integers from 1 to 100 in his mathematics lesson, and he did it within seconds. Can you calculate the sum as well without using a calculator? More clue: try using the method which was used by Gauss, where he broke down the numbers in pairs: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101 and so on. Answer: (4-digit number)

6. Joseph Lagrange, a Italian mathematician proved Bachet's conjecture in 1770. This conjecture is better known as Lagrange's four-square theorem, which states that every positive integer can be expressed as the sum of four squares. For example, 1 = 1^2 + 0^2 + 0^2 + 0^2, 2 = 1^2 + 1^2 + 0^2 + 0^2 and 3 = 1^2 + 1^2 + 1^2 + 0^2. Now, what is the value of these four squares, 8^2 + 9^2 + 11^2 + 20^2? More clues: this number is the Number of the Beast. Answer: (3-digit number)

7. This is yet another intriguing and interesting number. This number is the Kaprekar constant, after its discoverer, Dattaraya Kaprekar, an Indian mathematician. So, let's explore this number. Take any four-digit number (not all digits the same like 1111 and 2222) then construct the greatest and smallest numbers that can be formed from the four-digit number. After finding out their difference, repeat the same process for many times and eventually you will reach the Kaprekar constant. Do you know what the value of the Kaprekar constant is? More clues: the first two digits are three less than the square of 8 and the last two digits are seven less than the square of 9. Answer: (4-digit number)

8. Perhaps two of the most beautiful transcendental numbers (not the root of any polynomial functions) in mathematics are pi (3.142...) and e (2.718...). The former is also known as the Archimedes' number while the latter is better known as Euler's number. Leonhard Euler was a Swiss mathematician who introduced the notation of e (exponent) for the base of the natural logarithm (In). Euler was also credited with the introduction of the Greek letter pi in Euclidean geometry. Now, if you solve (e^pi) - pi the answer will approximate to which number? More clues: this number is a multiple of 10. Answer: (2-digit number)

9. You may have heard of the famous Fibonacci numbers and its sequence (0, 1, 1, 2, 3, 5, 8, 13... where the next number is the sum of its previous two numbers) that was introduced by Leonardo Fibonacci, a Italian mathematician. How about the Lucas number and its sequence? It was named after a French mathematician, Edouard Lucas. While the first two Fibonacci numbers are 0 and 1, the first two Lucas numbers are 2 and 1. So, the Lucas sequence is 2, 1, 3, 4, 7, 11, 18... Only 3 numbers occur in both the Fibonacci sequence and the Lucas sequence. What is the product of these 3 numbers? More clues: this number is a multiple of 3. Answer: (1-digit number)

10. This number's special property was discovered by Ramanujan, a great Indian mathematician. It is the smallest number that can be written as sum of two cubes in two different ways. More hints: the first two digits is the prime number after 13 and the last two digits is the prime number before 31. Answer: (4-digit number)