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# 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 chess, chess board, counting, rectangles, squares
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 approximants, approximation, floor, pi, sequence
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

## 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.

## 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

## 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