On Counting Things -- The "One-Off" Problem

We learn counting even before addition. It seems like it shouldn't be hard, but sometimes there is a trick to it. Sometimes people get it wrong, in ways that matter. I have a couple of examples in this post that I found amusing.

We'll be dealing with more advanced math than counting at this blogsite, for sure. I'm just getting started here. But it's revealing that even with what should be the simplest math, there can be confusion.

First, an explanation of the one-off problem, by example: Suppose that telephone cables span 1200 feet, supported by telephone poles every 100 feet. How many telephone poles are there? The knee-jerk answer is 12, which you would get by just dividing 1200 by 100. The correct answer though is 13 telephone poles. Without the 13th pole on the end, you would have the last 100 feet of telephone cable just dangling. The correct answer then is "one-off" from the answer that might come to mind immediately.

This one-off phenomenon can show up in what seem like different settings, and we all have to watch out for it.

In the late 1970s, I got a washing machine through a rent-to-own furniture and appliance shop. The agreement was that after 12 monthly rental payments, I would own the washer, with no further payments after that of course. (Maybe you can see where this is going. 😒) I got the washer in mid-July with my first monthly payment. I paid the monthly payments, mid-month of every month. I showed up back at the shop the next year in mid-June with what I saw as the last payment, and I asked the shop for the ownership papers that would acknowledge that I had made all the payments required for ownership. The young lady representing the shop was genuinely surprised by my request. She told me (in a tone suggesting that she thought this was just obvious) that, because my first payment had been in July, then naturally my last payment would have to be the July payment (the next year). I think her exact words were "July to July would be one year." (Obvious, right?) I disagreed. I was pretty sure I could count to 12 effectively.

So I was polite but insistent that I now did own the washing machine. The young lady... I'm tired of calling her "the young lady". Let's call her Eris, after the Greek goddess of chaos and confusion. Eris and I had some more back and forth about this, and then we agreed we should talk with the manager. Together the three of us ran over the details again. The manager: "But July to June is only 11 months. It was only 11 months ago that you started the contract. You've only made 11 payments." Do you see the one-off problem here? It's like the telephone poll situation, but in reverse: you do need the 13th telephone pole to sort of "end cap" the series of telephone cables, but you don't need a 13th monthly rental payment to "end cap" the 12 monthly rental periods. Whether or not you have to "add one" just depends on the situation. Yes, the 12th and last payment comes only 11 months after the first payment, but that's not relevant.

The manager and Eris clearly felt they were having to be very patient with me. We had some more back and forth ... July to June is only 11 months, blah blah blah. Then I brought out the big guns; I started counting on my fingers, refusing to be put off by the eyeball rolling of Eris and her manager. I did it real slow so they could see I wasn't pulling any fast ones. Finger number one was July of the earlier year. When I got to finger number 12, for 12 payments, saying "June" on that last count, you would have thought by Eris's and the manager's stunned expressions that I had just done some kind of magic trick. They both kind of paused, looking off into space, up and to the side, in the way body language experts say that people do when they're deep in thought processing information. Then they agreed I was right. As I drove home with my ownership papers, I wondered how many people they had successfully convinced to make a 13th monthly payment on a 12-month contract, through the power of their math.

With AI available now, we should be able avoid these kinds of confusions, right? Well, maybe not. I heard recently that a well known AI tool flubbed the following question: If it takes a carpenter 10 minutes to saw a board into 2 pieces, how many minutes does it take for the carpenter to saw it into 3 pieces? The AI just prorated the time based on the number of pieces; going from 2 pieces to 3 pieces is a 50% increase, so the AI answered 15 minutes for the 3 pieces -- 5 minutes per piece. The error though is focusing on pieces rather than on cuts. It's cuts that take time. To make 2 pieces, there is only 1 cut. To get 3 pieces, there are 2 cuts. Because it takes 10 minutes per cut, the answer for cutting into 3 pieces, is 20 minutes, not 15 minutes. After hearing this, I tested ChatGPT with this, and it did it correctly. By the way, there are some simplifying assumptions here: probably it's a rectangular board and each cut traverses the same extent and thickness of the board, and so on.

Some of the one-off problems involve the use of ordinal numbers. For example, suppose you pick up a prescription for a symptom on Monday, and you start taking it that day. Your doctor says it will take 3 days for the medicine to help. You're anxious for the symptom to go away, so you're disappointed when Wednesday morning arrives -- the 3rd day -- and you still have the symptom. But actually Monday to Wednesday isn't 3 days, it's 2 days. If you picked up the prescription late Monday afternoon, then Wednesday morning is actually just over 1.5 days later, not 3 days later. Monday should be counted as day 0, not day 1.

About 49 years ago I started working at the Law Library of the State of Iowa. We served mostly the Legislature, AG's office, and attorneys. Occasionally though we would help the lay public with legal questions, trying hard to avoid giving actual legal advice. One day a lady called to share with me her theory that everyone on Social Security has been shorted by about two monthly payments. To explain her theory for this, let's ignore leap years -- leap years only confuse the discussion more. Her theory relied on the notion that, because a year has 365 days, people actually turn one year old on the day before their traditional "first birthday". This, she said, is because the day before the traditional first birthday is the 365th day. So, people turn 2 years old 2 days before their traditional 2nd birthday, and so on. She reasoned that people turn 65 years old about 65 days before their 65th birthday. I won't belabor the explanation of the error here. I think we can all see it. "The 365th day" comes only 364 days after the first day. So people don't turn 65 years old 65 days before their 65th birthday. This is a one-off error compounded 65 times.

It can be hard to be right about complex issues. Sometimes even the things that should be simple can throw us off.

The image for this blog was AI generated. I asked Copilot to render in the style of Salvador Dali, but to keep it subtle. No drooping clocks. I think Copilot did pretty well.