## A Beautiful Puzzle

This lovely puzzle (for upper-elementary and beyond) is from Nikolay Bogdanov-Belsky’s 1895 painting “Mental Calculation. In Public School of S. A. Rachinsky.” Pat Ballew posted it on his blog On This Day in Math, in honor of the 365th day of the year.

I love the expressions on the boys’ faces. So many different ways to manifest hard thinking!

Here’s the question:

No calculator allowed. But you can talk it over with a friend, as the boys on the right are doing.

You can even use scratch paper, if you like.

And if you’d like a hint, you can figure out square numbers using this trick. Think of a square number made from rows of pennies.

Can you see how to make the next-bigger square?

Any square number, plus one more row and one more column, plus a penny for the corner, makes the next-bigger square.

So if you know that ten squared is one hundred, then:

… and so onward to your answer. If the Russian schoolboys could figure it out, then you can, too!

### Update

Simon Gregg (@Simon_Gregg) added this wonderful related puzzle for the new year:

## Beauty in Math: A Fable

Have you ever wondered what mathematicians mean when they talk about a “beautiful” math proof?

“Beauty in mathematics is seeing the truth without effort.”

“There’s something striking about the economy of the counselor’s construction. He drew a single line, and that totally changed one’s vision of the geometry involved.

“Very often, there’s a simple introduction of something that’s not logically within the framework of the question — and it can be very simple — and it utterly changes your view of what the question really is about.”

CREDITS: Castle photo (top) by Rachel Davis via Unsplash. “A Mathematical Fable” via YouTube. Story told by Barry Mazur. Animation by Pete McPartlan. Video by Brady Haran for Numberphile.

## Confession: I Am Not Good at Math

I want to tell you a story. Everyone likes a story, right? But at the heart of my story lies a confession that I am afraid will shock many readers.

People assume that because I teach math, blog about math, give advice about math on internet forums, and present workshops about teaching math — because I do all this, I must be good at math.

Apply logic to that statement.

The conclusion simply isn’t valid.

## A New Graph-It Puzzle

Since I’ve been posting new Alexandria Jones stories this week (beginning here), I’ve gone back and re-read the old Christmas posts. I noticed that the original Graph-It Game included a religious design, but nothing for those who don’t celebrate Christmas.

So I updated the post with a new, non-religious puzzle. Here it is, if you want to play…

### Graph-It Game Design

For this design, you will need graph paper with coordinates from −8 to +8 on both the x- and y-axis. Connect the points in each line. Stop at the periods, and then start a new line at the next point.

(-8,8) – (-8,0) – (0,8) – (-8,8) – (-4,4) – (0,4) – (0,8) – (8,8) – (4,4) – (0,8).

(8,8) – (8,0) – (4,0) – (4,-4) – (8,0) – (8,-8) – (0,-8) – (4,-4) – (0,-4) – (0,-8) – (-8,0) – (-8, -8) – (0,-8).

(-8,-8) – (4,4) – (0,4) – (4,0) – (4,4) – (8,0).

(8,-8) – (-4,4) – (-4,-4) – (0,-4) – (-4,0) – (-8,0).

(0,-2) – (0,-4) – (4,0) – (2,0) – (2,-2) – (-2,-2) – (-2,2) – (2,2) – (2,0) – (1,1) – (1,0) – (2,0) – (0,-2) – (-2,0) – (0,2) – (1,1) – (-1,1) – (-1,-1) – (1,-1) – (1,0) – (-4,0) – (0,4) – (0,-1) – (-1,0) – (0,1) – (1,0) – (0,-1) – (0,-2).

Color in your design and hang it up for the whole family to enjoy!

Of course, the fun of the Graph-It Game is to make up your own graphing puzzle. Can you create a coordinate design for your friends to draw?

### Want More?

You can see all the Alexandria Jones Christmas posts at a glance here:

“Love Christmas Lights” photo by Kristen Brasil via Flickr (CC BY 2.0).

## Review: Math & Magic in Wonderland

Are you looking for a fun book to read over the summer? I just finished Lilac Mohr’s delightful Math & Magic in Wonderland, and I loved it.

Highly recommended, for kids or adults!

A Jubjub bird disguised as a lark,
Borogroves concealing a snark,
When you’re in Tulgey Wood, you must
Be careful whom it is you trust…

With the discovery of Mrs. Magpie’s Manual of Magic for Mathematical Minds, Lulu and Elizabeth embark on an exciting journey to a realm inspired by Lewis Carroll’s poetry. The twins must use ingenuity and sagacity to solve classic logic puzzles that promise to uncover the book’s secrets and earn them The Vorpal Blade. In this interactive novel, the reader is invited to play along with the two heroines on their grand mathematical adventure.

Do you have the smarts to help Lulu and Elizabeth outwit the frumious Bandersnatch?

It’s time to enter Wonderland and find out!

–from the back cover of Math & Magic in Wonderland by Lilac Mohr

### What I Liked

Puns, poetry, and plenty of puzzles. Tangrams, tessellations, truth-tellers and liars. History tidbits and many classics of recreational mathematics.

The sisters Lulu and Elizabeth seem real — though perhaps more widely read than most of us. They are different from each other. They make mistakes and have disagreements. But they never deteriorate into the cliché of sibling rivalry that passes for characterization in too many children’s books.

In each chapter, the girls must solve a language, math, or logic puzzle to proceed along their journey. Then a “Play Along” section offers related puzzles for the reader to try.

No matter how challenging the topic, the book never talks down to the reader.

### What I Didn’t Like

… Um … Honestly, I can’t think of anything.

Since it’s traditional to criticize the editing of self-published books, I will say this: There was at least one place where the wording seemed a bit awkward. I would have phrased the sentence differently. But don’t ask me to identify the page — I was too caught up in the story to bother jotting down such a quibble. And I tried flipping through the book as I wrote this post, but I can’t find it again.

Unless you hate logic puzzles and despise Lewis Carroll’s poetry.

But for everyone else, this book is truly a gem. If you like The Cat in Numberland or The Man Who Counted, then I’m sure you’ll enjoy Math & Magic in Wonderland.

Disclaimer: Like almost all book links on my blog, the links in this post take you to Amazon.com, where you can read descriptions and reviews. I make a few cent’s worth of affiliate commission if you make a purchase — but nowhere near enough to influence my opinion about the book.

### And Now for the Giveaway

Lilac offered a paperback copy of Math & Magic in Wonderland for one lucky reader of Let’s Play Math blog.

The giveaway is done. Congratulations, Keshua!

• Tell us about your favorite language, math, or logic puzzle book! Or share a book you’ve been wanting to read.

## Noticing Fractions in a Sidewalk

My daughters didn’t want to admit to knowing me, when I stopped to take a picture of the sidewalk along a back street during our trip to Jeju. But aren’t those some wonderful fractions?

What do you see? What do you wonder?

Here is one of the relationships I noticed in the outer ring:

$\frac{4 \frac {2}{2}}{20} = \frac {1}{4}$

And this one’s a little trickier:

$\frac{1 \frac {1}{2}}{12} = \frac {1}{8}$

Can you find it in the picture?

Each square of the sidewalk is made from four smaller tiles, about 25 cm square, cut from lava rock. Some of the sidewalk tiles are cut from mostly-smooth rock, some bubbly, and some half-n-half.

I wonder how far we could go before we had to repeat a circle pattern?