In the car, while I was driving, my wife was finishing up the border of a quilt by hand. During the drive, I would keep asking how far along she was on finishing the border, which had four sides of equal length.
Sometime in the morning she says, “I’m about half way to a quarter, so an eighth.”
Later she says, “I’m a quarter of the way.”
Later she says something more interesting, “I’m half of the way, to the halfway,…”
“Huh?”, I inquire. “Isn’t halfway to halfway equal to a quarter of the way, which is where you were an hour ago?”
She adds, “No, Im half of the way, to the halfway from when I was at a quarter.”
I pause and think about this for a while, saying, “OK. From where you were before. You’ve made half of the progress to half based on being a quarter of the way…. That’s an interesting way to think about it. So, you are 3/8th of the whole way, right?”
She thinks about it for awhile, and says, yup.
She then want to explain how’s she been thinking about it, “I’ve been thinking about each side of the quilt as one. So, I was half way into completing the second side.”
“OK. I see. You were half way into the second 1/4,… which I think is why I said it’s an 1/8th, or rather the 3rd 8th.”
She thinks about that for a while and agrees.
So now, I’m curious about her way of thinking about it, and I ask, “Let me know when you get half of the way to the halfway from being half of the way to halfway from when you were at a quarter.”
She thinks about it for a while, stopping me several times from interrupting her,
After 20 seconds or so, she says, “So you want to know when I’m 7/16th of the way?”