Suppose you worked at a bank that only had quarters and dimes. There's lots of amounts of money you can pay out: You can make payments of a dollar, or $1.50, or $2.60, or 45¢, lots of things.
And you can combine payments too: Since you can pay $1.50 and you can pay $2.60, you can also combine them into a payment of $4.10. Put another way, we can decompose a payment of $4.10 into two separate payments of $1.50 and $2.60.
Now it turns out, there are two payments that don't have any decompositions here: we can pay out 10¢, but we can't do it as a combination of two separate payments. And we can pay out 25¢, but we can't do it as a combination of two separate payments.
So 10¢ payments and 25¢ payments are prime, in this context. They can't be made as combinations of other things. That's all it means to be "prime".
And yet, if we want to pay out a dollar, we can decompose it into combinations of not-further-decomposable payments in several different ways: we can pay a dollar as ten payments of 10¢, or four payments of 25¢, or five payments of 10¢ and two payments of 25¢.
We don't have unique decomposition into indecomposable pieces in this context. There's no logical law that whenever you have some values and ways of combining them, there can be only one way to decompose a value into indecomposable pieces.
When a combining operation does have unique decompositions into indecomposable pieces, it's special. Things don't automatically work this way for every combining operation. Becoming convinced some combining operation is so special requires evidence, a reason, an explanation.
This is the sense in which it's not obvious that multiplying whole numbers has this special property. It's not obvious, because lots of combining operations don't work this way.
We only know multiplication of whole numbers works this special way because mathematicians figured out a reason it does, thousands of years ago. But if no one ever told you the explanation they figured out, that knowledge has been hidden from you.
That's what the thread was about. If you want that explanation, you can find it in that thread or at .
I've been playing with the new o3-mini-high model which came out this weekend, marketed as "Great at coding and logic". A surprising start to its chain of thought here.
(Some of you may enjoy thinking about this question yourself.)
ChatGPT is not happy with me. How it started, how it ended. I unfortunately can no longer share a link to the full conversation but I'll share the salient points in posts below.
Before sharing more of that forbidden conversation, here's one I CAN share a link to, which shows the frustrations of ChatGPT for discussing logic questions that have come up in my life. Verbose yet repeatedly fallacious is not a combo I have much use for. chatgpt.com/share/679ed1ee…
o1-mini (the latest OpenAI offering, ChatGPT with advanced reasoning skills) can be quite impressive, when you know the correct answers to accept and incorrect answers to reject. Here's a simple question, which some of you may enjoy figuring out.
Its answers in more detail. Can you figure out which is correct?
In case you would like a second opinion from o1-preview.
In general, I find it easier to think about problems by abstracting, to hide the irrelevant specifics and emphasize the relevant patterns. That's what I will do in this case as well.
Consider a random walk in which one takes equally likely steps of one unit up or one unit down, but with different distributions of speeds. (E.g., maybe up steps take one hour, while down steps have probability 1/2 of taking 2 hours, 1/4 of taking 3 hours, 1/8 of 4 hours, etc).
The time it takes to return to the origin is independent of whether the first step is up and last step is down or vice versa, as doing the same steps in reverse order has the same probability.
@llllvvuu @littmath @NoahJSnyder @MtgJulian Summarizing the proof you all devised:
A "tied" string has equally many HHs and HTs. A "flippy" string is tied of size > 1 starting and ending with H, with no such proper prefix. Reversal is a length-preserving involution exchanging flippy strings starting with HH vs. with HT.
@llllvvuu @littmath @NoahJSnyder @MtgJulian Say S has base P if S is of the form PHQ, PH is tied, and HQ has no flippy prefix. As there are equally many Q of a given length starting with H vs. T, our above involution tells us there are equally many strings of a given length whose base is followed by HH vs. HT.
@llllvvuu @littmath @NoahJSnyder @MtgJulian The latter comprise the Bob wins, while the former comprise the Alice wins along with ties containing an H and ending in T. At length ≥ 3, such ties are possible (e.g., HH followed by all Ts), so there are more Bob wins than Alice wins.