1) Spring used to be the beginning of our calendar. This practice antedated Christianity, and reflects the rebirth of the sun or renewal of life. The Church adopted the custom, and for many centuries our ‘year’ began with easter. We mark this rotation back to the beginning with a simple and classical question, namely how to calculate the sum of the first x natural numbers without adding them all up.
2) Our first question involved the triangular numbers, which are 1, 1+2, 1+2+3, 1+2+3+4, etc. What digits can never appear at the end of a triangular number? What is the only triangular number that is a prime? Can you find a natural number that can’t be represented as the sum of 3 or less triangular numbers?
The answers to last month’s puzzles were supplied last month.
Here are the answers to this month’s puzzles:
1) Gauss gave us the answer, which is (x/2)(y + z), where x is the number of numbers being added, y is the first number, and z is the last. The usual story is that Gauss discovered this method while solving a problem set by his teacher in elementary school, and that the numbers were 1 through 100. Gauss himself, however, said that his teacher had given the class a particular five-digit number and asked the students to add a particular three-digit number to it 100 times in succession, then find the sum of that series. Does the above method work if the story is changed as we’ve suggested?
2) The digits that can never appear at the end of a triangular number are 2, 4, 7, or 9. The only triangular number that is a prime is 3. All natural numbers can be represented as the sum of 3 or fewer triangular numbers (discovered by our friend Gauss).
[This month’s puzzles are adapted from Posamentier and Lehmann’s Mathematical Amazements and Surprises]