This week’s challenge is an original suggested by Naoki Inaba, and is the second in a series by this ingenuous and remarkably prolific Japanese puzzle designer. (In case you missed it, check out *Predator and Prey*.)
This week we take on Menseki Meiro, or “Area Maze,” referring to the labyrinthine process involved in finding the target — the area of a rectangle. Area Maze is already
a hit in Japan. Will it have the same addictive potential in the United States as it does in Japan? Let’s give it a try. Here’s Mr. Inaba presenting five challenges in ascending
difficulty:

Menseki Meiro (Area Maze)

This puzzle is so simple as to need no explanation: Figure out what is the value of the ‘?’.

(The figures in the questions are not always drawn proportionally accurately so that it cannot be solved by guesswork.)

Mr. Inaba introduces this fifth challenge with the following note:

The problems I sent are too easy as a challenge. So I attached a big size problem [shown below].

Mr. Inaba continues with a bit about the origin of the puzzle and a suggestion for solving:

This puzzle was originally designed as teaching material for elementary school children at the request of a head of a cram school. Therefore, it is made to be able to be solved without decimals or fractions, for the children who have just learned to calculate the area of rectangles. Of course, you can crunch the number with decimals and/or fractions. But you should throw away the need for extravagant calculations, and manage to set up equations using integers only. This simplicity makes it an interesting puzzle.

Solving a mathematics problem, people usually focus on what the answer is. As for this puzzle, the integer-only restriction makes it fun to follow the path to the answer, as if you were navigating a maze.

Now this type of puzzle is now appearing in some puzzle magazines and people of all ages are enjoying them. Some puzzle books were published, and even an app named “Area Maze” became available. [The app is available for both iOS and Android.]

Thank you, Mr. Inaba. With that we wrap up this week’s introduction to Area Maze. As always, once you’re able to read comments for this post, use Gary Hewitt’s Enhancer to correctly view formulas and graphics. (Click here for an intro.) And send your favorite puzzles
to gary.antonick@NYTimes.com.

**Solutions**

We’ll start with the area of the largest maze. Here’s Mr. Inaba:

The answer is 100cm².

A and B:Use the sum of the areas of each two shaded rectangles. When you find the width of rectangle C, you can also find the width of rectangle D. The height of rectangle E can be found by using addition and subtraction. In this problem, the whole picture itself is a rectangle, so the heights of the both sides are the same, and you can find the height of rectangle X. In the same way, you can find the width of rectangle X.

As for the introductory four challenges — here’s **Ravi**:

The first four took less than a couple of minutes but the harder one took about an hour and I loved the way when I felt I was stuck, there was some way to get going again. It was beautifully constructed to ensure that you have to solve nearly every rectangle whose area is given, before you got the size of the shaded area.

The answers I got the first four going clockwise are 16 cm², 30 cm², 50 cm² and 6 cm² respectively.

The fifth one’s answer is a square of size 100 cm².

Instead of finding every rectangle, I tried finding the overall square dimensions and then tried to get the dimensions of the shaded square. Lines that do not cut across more than one rectangle are not determinable unless the other side of the rectangle is known. So focused on those first.

Dr W shared the following graphic:

Dr W also tried pushing into other shapes. Here was his initial attempt with triangles, solved by **Ravi** as follows: “It is clear that the areas are all squares. The bigger triangle is 144,
the triangle at the right bottom is 64, the triangle at the top is 16 and the smaller triangle is 1.”

Mr. Inabi followed with this additional challenge in 3D:

Thank you, Mr. Inabi, and thanks as well to everyone who participated this week: Ravi, Dr W, Daniel, Seth Cohen and LAN.