Category Archives: solutions

Chessboard counting: solutions and further challenges

And now for some solutions to the chessboard counting challenges. The first challenge was to count the number of squares of any size on an 8×8 chessboard. The key here (as with many counting problems) is to break the problem … Continue reading

Posted in challenges, counting, solutions | Tagged , , , , | 2 Comments

Predicting Pi: solution

Now for the solution to the question in my previous post, which asked what you can learn about , given the sequence of integers . Nick Johnson commented: Well, the obvious thing one can learn given just |(10^n)r| is the … Continue reading

Posted in convergence, pattern, sequences, solutions | Tagged , , , , | 5 Comments

Challenge #12 solution, part III

And now for the solution to problem #3 from Challenge #12, which asked: how many ways are there to write a positive integer n as a sum of powers of two, with no restrictions on how many powers of two … Continue reading

Posted in counting, pattern, recursion, sequences, solutions | 1 Comment

Challenge #12 solution, part I

I’ll begin by providing an answer to the first of the three questions I posed in a previous post.

Posted in counting, pattern, proof, solutions | Tagged | 8 Comments

Perfect numbers, part III

This is the last in a series of posts about perfect numbers. A quick recap of the series so far: in part I, I defined perfect numbers as positive integers n for which , where denotes the sum of the … Continue reading

Posted in algebra, number theory, open problems, primes, solutions | 11 Comments

Nuclear Pennies Game: Solution

It seems that everyone with a blog is always apologizing for not posting in a while, as if this has somehow inconvenienced their readers. Of course, with the magic of feed readers, email notifications, and the like, this is not … Continue reading

Posted in games, links, solutions | Tagged , , , | 1 Comment

Challenge #10 Solution

Have you tried solving Challenge #10 yet? Go try it first if you haven’t. It’s not too hard, I promise!

Posted in famous numbers, golden ratio, proof, solutions | 3 Comments