Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.

Alexandria Jones and the Christmas Present Quandary

Magic Square Puzzles

## How Many Tessellating Shapes Can You Find?

Any triangle (regular or scalene, obtuse or acute) will tessellate, as will any quadrilateral. Regular hexagons can make an infinite honeycomb pattern. Many combination shapes will also tessellate, such as the “Square+Triangle=House” shape shown here.

## Lo-Shu: The Turtle’s Design

The shapes on Lo-shu’s back stand for beads or knots on a string, and Leon counted each set of beads to find the numbers of the magic square. The white beads were for odd numbers; the black beads were even.

The other set of multigrades in the magic square were the side columns:
{3, 4, 8} and {2, 6, 7}.

## Leon’s Magic Square Puzzle

Alex decided that Leon’s “perfect” solution had to contain the perfect number 6. Using the clues Leon had already given, she was able to eliminate every other possible combination.

## And a Perfect Challenge

6 is called a perfect number because it is equal to the sum of its divisors. 1, 2, and 3 will all divide evenly into 6, and $1 + 2 + 3 = 6$.

Of course, 6 will also divide evenly into itself, but if we counted the number itself, the sum would always be too big. So we say that a perfect number is any number whose proper divisors — that is, all the divisors less than the number itself — add up to make that number.

Perfect numbers are rare, but there is another one that’s less than 100. Most numbers are either deficient (the sum of the divisors is less than the number) or abundant (the sum is greater than the number).

Can you find another perfect number?

Do you think there are more abundant numbers, or more deficient ones?

Which category do prime numbers fall under? What about square numbers? Odd numbers? Even ones?…

Of course, you can find answers to these challenge questions on the Internet (Number Gossip). But if you explore the numbers for yourself, you will build your factoring skills and discover some interesting things along the way.

## To Be Continued…

Read all the posts from the November/December 1998 issue of my Mathematical Adventures of Alexandria Jones newsletter.