(Step 15, try to get a picture...)
First up, if you don't know how to do the box braid, I found this tutorial
really helpful. But there are many many out there.
My biggest problem is that after a bit (hours...) my hands hurt from trying to hold onto the knot. I thought a loop would be easier to hold on to, and so I tried to figure out a way to do it that was (somewhat) strong and would stand up (to tugging - the chewing... well...)
Step 1 - cut your strips. These don't need to be perfect, mine are roughly 2" wide.
Step 2 - take four strips, two of each color, and tie a loose knot in the center.
Step 3 - start braiding out from the knot.
Step 4 - braid from the knot to close to the end (but leave a bit!)
Step 5 - repeat step 3 in the other direction.
Step 6 - pause to get help from someone mocking you with the one she's already destroyed.
Step 7 - lay it out and "measure" the handles. Note - it is very stretchy, so you can probably make the loops smaller than you think. I am using only one edge-to-edge width of fleece for mine, you can sew two strips end-to-end if you want a longer tug.
Step 8 - pull one of the ends through the to opposite side of the already braided bit. I use a big crochet hoop and snip a hole in the very end of the strip to pull.
Step 9 - pull the "front" and "back" strips through to the "front" and "back". Note that the red ends are on the red sides and the blue ends are coming out the blue sides.
Step 10 - this is super hard to photograph, but what you do is flip it over (the loop is down in the picture) and you're going to repeat the same box braid OVER the box braid in the "center" of the tug.
Step 11 - like this...
Step 12 - so it looks like this...
Step 13 - do both ends this way, and when they meet in the middle...
Step 14 - knot them together (square knots, not granny knots... and if you leave a little gap the knots will kinda nestle in... you can be all fancy and weave in the ends and make it pretty, but my dog destroys stuff in about 10 minutes flat, so I just don't care).
Step 15 - attempt to get a picture while someone is trying to grab the new toy...
Step 16 - give up and play.
Dog tested. Dog Approved. Shortly to be Dog Destroyed, I'm sure.