For over a thousand years, people have been drawing intricate woven patterns known as Celtic knots. As we’re about to find out, there’s a lot of maths involved!

- Pencil and eraser
- Pen
- Diagonal grid paper

- Look at the grid paper. It’s similar to normal grid paper, but lined up on the diagonal. Look at the top-left corner and follow the line of squares going down-right. Take your pencil and gently shade the line of squares directly above the corner line. Also shade the line of squares directly below.
- You have two shaded rows separated by a blank row. Carefully shade every second row in the grid in the same way.
- Look at the top right corner and follow the row of squares running down-left from that box. Shade the rows directly above and below this row, and then, as before, shade every second row running in that direction. The shaded rows indicate where the ribbons will run.
- The next step is to draw the sides of the ribbons. Look at the top-left corner. The two ribbons (shaded rows) running down-right here need to join together to make one loop. Draw a little curve on the inside and a big curve on the outside to join the two ribbons into one looped ribbon.

- The two ends of the loop now encounter a shaded row running across them. One end of the loop will go over the row – draw the sides of the ribbon in. The other end of the loop will go under the shaded row – so draw the sides of the row over the ribbon.

- Follow to the next shaded row. If the ribbon went over last time, this time it goes under. If it went under, this time it goes over. Keep filling in all the intersections using this over-under pattern.
- When you get to the edge of the grid, curve the ribbon to join a row running in the other direction.

- If your ribbon gets to a corner, make the same sort of pattern you did at the first corner.
- Once you’ve traced the ribbon all the way back, you have finished your Celtic knot. It is all one twisted path that goes over, under, over, under the whole way around.
- To neaten everything up, check for mistakes, then go over the lines with pen. Finally, erase the pencil shading.

This activity has you drawing big tangled Celtic knots. But if you follow a ribbon all the way around, you’ll notice that it’s all one big loop! This isn’t always the case – if you start out with a square instead of a rectangle, you’ll end up with more than one loop. Having a rectangle isn’t enough either.

In this activity, there are four shaded rows going in the same direction into each short side. But there are five leading into each long side. This difference means that one ribbon will end up going on different rows.

If the rectangle had four and six rows on the long and short sides, you could follow a ribbon going onto different rows too. But if you picked an even row – maybe the second one – that ribbon would only visit even rows. If you picked an odd starting point, you would only visit odd rows. If a ribbon only visits even rows, or odd rows, more than one loop is needed to make the knot.

In order to have only one loop, the number of rows in long and short sides must not share factors. That means that no number will evenly divide the number of rows in short side as well as the long side.

Another tangled activity, making an impossible plait

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