## Diagonals on a grid

By David Shaw, 18 March 2020

Moein is messing about with some pieces of graph paper. He’s cut out some rectangles, and he’s ruling a line from the top left corner to the bottom right corner of each piece of paper. Then he plans to count the number of squares that each diagonal goes through.

**Difficulty: Easy**

First, Moein is going to try a grid that’s five squares wide and five squares tall. How many squares does the diagonal go through?

**Difficulty: Tricky**

The next grid is five squares wide and four squares tall. How many squares does the diagonal go through this time?

**Difficulty: Taxing**

Finally, there’s a grid that’s six squares wide and four squares tall. How many squares does this diagonal go through?

**Scroll down or click for a hint, or the answer!**

## Brainteaser hint

It might help to try some smaller examples first.

• With a grid that’s two squares wide and two squares tall, the diagonal goes through two squares.

• With a grid that’s two squares wide and three squares tall, the diagonal goes through four squares.

If all else fails, you could get some graph paper and do it yourself!

## Brainteaser answer

**Five squares wide and five squares tall**

The diagonal passes through the middle of the five squares on the diagonal of the grid! So the answer is five.

**Five squares wide and four squares tall**

As the diagonal moves from one square to the next, it has three options.

• The line can go out the side of the square into a new column.

• The line go out the bottom of the square into a new row.

• The line go out the bottom corner of the square into a new column and row at the same time.

The diagonal goes into a new column four times and into a new row three times. Each time it goes through an edge. This line will not touch any corners except the one it starts on and the one it ends on. The diagonal starts in a square and crosses seven edges, which means it’ll touch eight squares in total.

**Six squares wide and four squares tall**

With this grid, you can use the same strategy as the previous grid. But there’s one big difference. The diagonal will pass through the middle of the paper, which is a corner. This is the only corner it will go through.

The diagonal starts in a square and goes into a new column five times and a new row three times. If these were all edge crossings, that would make nine squares. But the diagonal crosses one row and column at the same time, so this line only visits eight squares!

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20 March, 2020 at 3:52 pm

This reminds me of a puzzle my uncle gave to me 50 years ago. I found the puzzle too difficult, but when my uncle showed me the method, I realised the solution was easy to find (but fiddly).

The puzzle: There is a rectangular room 30 units long, 12 units wide and 12 units high. A huntman spider is in the centre of the end wall, one unit up from the floor. A fly sleeps in the centre of the opposite end wall, one unit down from the ceiling. What is the shortest distance the spider can run in order to catch the fly?

Solution: My uncle cut from a sheet of graph paper a diagram that, when folded, it ‘created’ the room. He cut it in such a way that a straight line could be drawn between the two points. Then he measured the line! 40 units.

I’ll try to include a photo:

/Users/markavery/Desktop/IMG_0716.JPG