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A Strange Grid Reveals Hidden Connections Between Simple Numbers

quantamagazine.org

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Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#21

Post by metrognome »

desc wrote: post #11

My intuition was already screaming 'it'll be the other way around for a geometric progression' before I read that far, but I'm damned if I can understand or even speculate why. Most likely I don't properly understand it. I'm inclined to wonder if there's a third operator which could be tested like this, such as exponentiation, but that's not commutative over integers. Of course, if there are similarly intriguing patt…

It's not too surprising that there are fewer unique products for a geometric progression, because subsequent terms in a geometric progression are found by multiplying by the common ratio. There will be fewer unique products when all of the numbers share that common ratio.

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#22

Post by Dylan16807 »

pwaai wrote: post #8

i dont get it...what is the hidden connection exactly?

'connection' is a somewhat misleading way of stating it. It's more like the opposite of a connection. A set of numbers cannot have trait A and trait B at the same time.

Trait A is that the numbers have a certain kind of similarity in their spacing, leading to few unique sums. Trait B is that they have a certain kind of similarity in their factors, leading to few unique products.

So from some angle it's interesting that you can't impose both types of patterns on a single set of numbers, because that inability implies a relation between two very different operations.

But on the other hand why would you assume that you should find an intersection between two restrictive algorithms? Maybe there's a hidden assumption that the algorithms would be random-ish and let you find an overlap if you search hard enough, and that assumption is screwing with people's intuition.

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#23

Post by Dylan16807 »

clairity wrote: post #3
CamperBob2 wrote: post #2

Maybe I don't understand the full implications of the problem, but it seems obvious that a multiplicative table will result in more 'numerical diversity' than a summation table. Both multiplication and addition yield points on lines described by the usual mx+b equation. Addition constrains x to 1, so all of the results will be clustered near the origin, with more coincidences as an unavoidable result. Conversely, if…

dunno, but it's neat to think about. multiplication can cover more ground in integer space than addition, so it can reach more unique places than addition for a given set of numbers. it's kinda like comparing how many squares knights and pawns can reach on a chessboard for a given number of moves.

If you give well-spaced source numbers, they can reach the same number of spaces.

For products it's easy: distinct prime numbers.

For addition you can pick numbers so that each one is at least [previous number] x [number of moves] + 1. To keep it extra simple go with [number of moves + 1]^n.

So this is very easy if the number of moves is known upfront, and impossible if it's not.

And in the source puzzle the number of moves is always 2.

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#24

Post by function_seven »

Dylan16807 wrote: post #22
pwaai wrote: post #8

i dont get it...what is the hidden connection exactly?

'connection' is a somewhat misleading way of stating it. It's more like the opposite of a connection. A set of numbers cannot have trait A and trait B at the same time. Trait A is that the numbers have a certain kind of similarity in their spacing, leading to few unique sums. Trait B is that they have a certain kind of similarity in their factors, leading to few unique products. So from some angle it's interesting th…

> Trait A is that the numbers have a certain kind of similarity in their spacing, leading to few unique sums. Trait B is that they have a certain kind of similarity in their factors, leading to few unique products.

This is an excellent way to describe it. I had the sense that “of course the geometric sequences will collide more when multiplying” but if you asked me “why?” I would have trouble explaining. So thanks.

As far as finding a set that has an equal number of distinct sums and products, would it be possible to come up with a sequence that’s “half”-arithmetic and “half”-geometric?

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#26

Post by feanaro »

Dylan16807 wrote: post #22
pwaai wrote: post #8

i dont get it...what is the hidden connection exactly?

'connection' is a somewhat misleading way of stating it. It's more like the opposite of a connection. A set of numbers cannot have trait A and trait B at the same time. Trait A is that the numbers have a certain kind of similarity in their spacing, leading to few unique sums. Trait B is that they have a certain kind of similarity in their factors, leading to few unique products. So from some angle it's interesting th…

> But on the other hand why would you assume that you should find an intersection between two restrictive algorithms?

I think it's key here that these two algorithms are so basic and fundamental to the structure they are executed upon (the naturals) so that every other algorithm will use them in some way. The very notion of "algorithm over the naturals" and intuition of what it means for such an algorithm to be restrictive are determined by the nature of these operations.

Put like this, it seems far less obvious to me that there is an a priori reason why the results of these two operations should or shouldn't coincide. Both scenarios seem plausible before we study problems like these.

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#27

Post by vanderZwan »

IshKebab wrote: post #25

Good job they explained what "distinct" means...

This is maths we are talking about, you should never assume that the common usage of words applies to the precise definition used in the maths context. Take the Boolean inclusive "or", for example.

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#28

Post by IshKebab »

vanderZwan wrote: post #27
IshKebab wrote: post #25

Good job they explained what "distinct" means...

This is maths we are talking about, you should never assume that the common usage of words applies to the precise definition used in the maths context. Take the Boolean inclusive "or", for example.

I think people that know 'product', 'geometric series', etc. know what distinct means. It doesn't have some special mathematical meaning here.

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#29

Post by Dylan16807 »

function_seven wrote: post #24
Dylan16807 wrote: post #22

Earlier quoted context omitted.

'connection' is a somewhat misleading way of stating it. It's more like the opposite of a connection. A set of numbers cannot have trait A and trait B at the same time. Trait A is that the numbers have a certain kind of similarity in their spacing, leading to few unique sums. Trait B is that they have a certain kind of similarity in their factors, leading to few unique products. So from some angle it's interesting th…

> Trait A is that the numbers have a certain kind of similarity in their spacing, leading to few unique sums. Trait B is that they have a certain kind of similarity in their factors, leading to few unique products. This is an excellent way to describe it. I had the sense that “of course the geometric sequences will collide more when multiplying” but if you asked me “why?” I would have trouble explaining. So thanks. A…

Even a single value that doesn't fit in one sequence will cause a lot of extra values, so your ability to fit multiple patterns gets worse and worse as you go longer. Something like 2 4 6 8 16 does pretty well for its size but even then it's giving you twelve distinct outputs for either operation.
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