A MATHEMATICIAN AT PLAY| Children

# Lose yourself in a maze of numbers

When I was about 10, I got into making mazes. I used to draw elaborate and curvy mazes with branchings and dead ends. I even contributed one to the kids section of my local newspaper, and got it published!

I still find mazes fascinating, but I’ve gotten into stranger variants since then. Today I wanted to share a few arithmetic mazes. For each, you are to try and end with the largest value possible.

In this addition maze, there are numbers to add in each cell. Starting at 0, you move from cell to cell, and build a larger and larger sum. When you get to “End,” you are done. You aren’t allowed to pass through the same cell twice. What is the largest sum you can get?

## Puzzle 2. Mixed Maze

Use the same idea as in the Addition Maze, except that now you can add or subtract. What is the largest total you can get?

## Puzzle 3. Mixed Maze

Use the same idea, except that now you can add, subtract, or multiply. What is the largest total you can get?

## Solution

One key to solving these mazes it to reduce them to their essentials by turning them into a graph. A graph is a collection of points and connections that can contain information. Once you begin a passage of the maze you have to continue until you get to the end. That means we can reduce the entire passage to a single move. It’s kind of like drawing a subway map. It’s not the same as the territory, but it abstracts it in a useful way.

For example, if you go from “Start” along the top passage in the first puzzle, you’ll go through eight 5s and one 4. That’s a total addition of 40 for the edge and 4 more for the final vertex. Continuing on in that fashion, we can redraw the entire map.

For me, the graph is easier to work with. In particular, it’s easier to compare different pathways. I’ll draw in the best one in red. It’s not too hard to see that any change would decrease your sum. Our total for this path is:

40 + 4 + 20 + 6 + 6 + 6 + 28 + 8 + 24 + 6 + 6 + 18 + 6 = 178. That’s the best you can do!

You can attach the next two mazes in the same way, though the third puzzle requires a bit more delicacy.

For Puzzle 2, my best score is 63. Here is how you can get that:

For Puzzle 3, the best score is 3609. Here is how you can get that:

Happy puzzling!

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