Problems

This rudimentary page is just a beginning, a work in progress.

How many rectangles (squares included) are on a standard 8-by-8 chess board?
Think About: Ask the same question with the number 8 replaced by the numbers 1, 2, 3, and 4. Can you guess a pattern? Does your pattern continue for 5? Can you see why the pattern holds for chessboards of all sizes?

How does one sum the first n positive integers? The first n positive squares? The first n positive cubes?
Think About: Can you guess an answer by trying n=1, 2, 3, and 4? If so, can you explain why your guess is correct for all values of n. If you knew that the guess was correct for n=100, could you argue why the guess would have to be correct for n=101?

Write down all the numbers from 1 to 100 on a piece of paper. Now pick two of these numbers, call them x and y, erase them, and write down the single number x+y+xy in place of them. Do this repeatedly until only one number remains. What number could that last number be?
Think About: What if you write down just 1, 2, and 3 on the piece of paper to start? What if you replace the formula x+y+xy with the simpler formula x+y? What if you replace the formula with xy?