Date: 23/03/2023 09:20:10
From: SCIENCE
ID: 2011301
Subject: Discrete Mathematics

Here we go, drawing inspiration (see below) and encouragement from The Rev Dodgson at https://tokyo3.org/forums/holiday/posts/2010843/ in thread https://tokyo3.org/forums/holiday/topics/15739ABC Argument teaser for pedants” started by ChrispenEvan.

First, the simple number problem that got people working, followed by a few clips of the answer(s), and then in subsequent posts we drop a little bit of code to help everyone join in the fun if they so desire.

https://tokyo3.org/forums/holiday/posts/2005858/

Reply Quote

Date: 23/03/2023 09:20:30
From: SCIENCE
ID: 2011302
Subject: re: Discrete Mathematics

The Rev Dodgson said:

The Rev Dodgson said:

Not from ABC, but here’s a simple little teaser for you all:

If a, b, and c are integers greater than 0, find values such that:

a/(b+c) + b/(a+c) + c/(a+b) = 4

I’ve just realised that is literally an abc argument teaser.

(all right, let’s begin)

Reply Quote

Date: 23/03/2023 09:21:15
From: SCIENCE
ID: 2011303
Subject: re: Discrete Mathematics

The Rev Dodgson said:

SCIENCE said:

The Rev Dodgson said:

Spiny Norman said:

The Rev Dodgson said:

Spiny Norman said:

The Rev Dodgson said:

Not from ABC, but here’s a simple little teaser for you all:

If a, b, and c are integers greater than 0, find values such that:

a/(b+c) + b/(a+c) + c/(a+b) = 4

https://www.wolframalpha.com/input?i=a%2F%28b%2Bc%29+%2B+b%2F%28a%2Bc%29+%2B+c%2F%28a%2Bb%29+%3D+4

Best I can do.

Scratches head.

So what is the answer?

NFI sorry. If I really had to solve it, I’d put the equation into Excel and just try various small numbers to try to get 4.

That’s what I tried but gave up eventually, fully expecting the solution to be bleeding obvious, once it had been revealed.

It’s actually not that easy. In fact the 3 lowest integers that satisfy the equation are 79, 80 and 81 digits long.

someone showed us the challenge before and we similarly gave up but we still didn’t understand what they were trying to metaprove

If you assume A = 1, I have come up with a relatively simple way of finding C for any selected value of B, but how you would go about finding the 3 lowest values for A,B,C that are positive integers, I still have no idea.

(…)

Reply Quote

Date: 23/03/2023 09:21:38
From: SCIENCE
ID: 2011304
Subject: re: Discrete Mathematics

JudgeMental said:

The Rev Dodgson said:

JudgeMental said:

The Rev Dodgson said:

Witty Rejoinder said:

How? That’s easy: ask Btm.

Yeah, I hope he has a look in.

DV is a maths gru.

OK, well dv goes without saying.

Now you tell me after I’ve said it!

(…)

Reply Quote

Date: 23/03/2023 09:22:04
From: SCIENCE
ID: 2011305
Subject: re: Discrete Mathematics

Spiny Norman said:

The Rev Dodgson said:

Spiny Norman said:

I thought I’d try ChatGPT with that problem.
(Apologies for the poor formatting)

It didn’t go as well as I’d hoped ->

Me: I’d like some help with a mathematical problem please.

GPT: Sure, I’d be happy to help you with a mathematical problem! What’s the problem you’re working on?

Me: What 3 positive integers satisfy the equation: a/(b+c) + b/(a+c) + c/(a+b) = 4

GPT: We can begin by simplifying the left-hand side of the equation and manipulating it algebraically.

a/(b+c) + b/(a+c) + c/(a+b)

= (a^2 + ab + ac + b^2 + bc + ac + ab + bc)/(a+b)(b+c)(a+c)

= (a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)/(a+b)(b+c)(a+c)

Now we can use the fact that the equation equals 4 to write:

(a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)/(a+b)(b+c)(a+c) = 4

Multiplying both sides by (a+b)(b+c)(a+c), we get:

(a^2 + b^2 + c^2 + 2ab + 2ac + 2bc) = 4(a+b)(b+c)(a+c)

Expanding the right-hand side, we get:

4(a+b)(b+c)(a+c) = 4a^2b + 4ab^2 + 4a^2c + 8abc + 4b^2c + 4ac^2 + 4bc^2 + 4abc

Simplifying this expression, we get:

4(a+b)(b+c)(a+c) = 4(a+b+c)(ab+ac+bc)

Dividing both sides by 4 and canceling (a+b+c) from both sides, we get:

(a+b)(b+c)(a+c) = (ab+ac+bc)

Now, let’s look at the equation more carefully. We see that the left-hand side is the product of three factors, each of which must be a factor of (ab+ac+bc). In other words, we can write:

a+b = k(ab+ac+bc)
b+c = l(ab+ac+bc)
a+c = m(ab+ac+bc)

where k, l, and m are positive integers.

Multiplying these equations together, we get:

(a+b)(b+c)(a+c) = klm(ab+ac+bc)^3

Substituting the equation we derived earlier, we can rewrite this as:

(ab+ac+bc) = klm(ab+ac+bc)^3

If ab+ac+bc is not equal to 0 or 1, we can simplify this expression to:

1 = klm(ab+ac+bc)^2

Since k, l, m, a, b, and c are positive integers, the only solution to this equation is ab+ac+bc = 1.

Substituting this value into the equation we derived earlier, we get:

(a+b)(b+c)(a+c) = 1

Since a, b, and c are positive integers, the only way this equation can be satisfied is if a = b = c = 1.

Therefore, the solution to the equation a/(b+c) + b/(a+c) + c/(a+b) = 4 is a = b = c = 1.

Me: Thanks, but it isn’t correct. Your solution would be 1/2 + 1/2 + 1/2 = 4

GPT: I apologize for my mistake. Let me try to solve the problem again.

First, we can simplify the expression on the left-hand side by finding a common denominator:

a/(b+c) + b/(a+c) + c/(a+b) = (a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)/(a+b)(b+c)(a+c)

Now, let’s try to simplify this expression further. We know that a, b, and c are positive integers, so we can assume that a <= b <= c.

Therefore, we have:

a^2 + b^2 + c^2 >= 3a^2

2ab + 2ac + 2bc >= 6ab

Putting these together, we get:

(a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)/(a+b)(b+c)(a+c) >= (3a^2 + 6ab)/(a+b)(b+c)(a+c)

= 3a/(a+c) + 6b/(a+b)(b+c)

We want to find three positive integers a, b, and c such that the expression above is equal to 4. So we have:

3a/(a+c) + 6b/(a+b)(b+c) = 4

Multiplying both sides by (a+c)(a+b)(b+c), we get:

3a(a+b)(b+c) + 6b(a+c)(b+c) = 4(a+b)(b+c)(a+c)

Expanding both sides, we get:

3a^2b + 3ab^2 + 6abc + 3a^2c + 6abc + 3b^2c = 4a^2b + 4ab^2 + 4abc + 4a^2c + 4b^2c + 4abc

Simplifying, we get:

a^2b + ab^2 + a^2c + 2abc + b^2c = 0

Since a, b, and c are positive integers, this equation can only be true if one of the variables is equal to zero. However, that would mean that the left-hand side of the original equation is undefined, so there are no solutions to the equation.

I thanked it and left that chat.

I’m pretty sure it made a mistake there somewhere.

It certainly made a whopper in the first attempt.

(…)

Reply Quote

Date: 23/03/2023 09:22:33
From: SCIENCE
ID: 2011306
Subject: re: Discrete Mathematics

roughbarked said:

roughbarked said:

Tamb said:

roughbarked said:

The Rev Dodgson said:

roughbarked said:

The Rev Dodgson said:

roughbarked said:

SCIENCE said:

The Rev Dodgson said:

dv said:

SCIENCE said:

dv said:

JudgeMental said:

dv said:

dv said:

The Rev Dodgson said:

OK, well dv goes without saying.

Sorry to disappoint my adoring fans but this has me stymied.

I’m leaning towards saying there’s no such solution but then we are left with a need to prove that.

It is kind of a scalable problem. Like if you have a solution a,b,c then ka, kb, kc is a solution. Therefore if there is a solution, there is a solution in which a b and c do not share a common factor.

Do they publish the answers for such teasers eventually?

not for the ones I post. they let the participants fight it out. the one the rev posted he just made up or got it from somewhere.

I hope this hasn’t been some wild goose chase

the Anser is out there

Amusing

What do geese have to do with it?

They’re a tough crowd.

sorry, nostra ruber clupea

what did you want me to do with your red herrings?

Thanks roughie, saved me a Binge :)

Just remember that clupea is herring.

So “mea clupea” means I am a herring?

Makes sense.

you are one ell of a speller.


Hair rings (Some of them red)

Clupea is genus of planktivorous bony fish belonging to the family Clupeidae, commonly known as herrings.
It also works for pilchard.

The red ones are for the rangas.

(…)

Reply Quote

Date: 23/03/2023 09:22:58
From: SCIENCE
ID: 2011307
Subject: re: Discrete Mathematics

SCIENCE said:

The Rev Dodgson said:

The Rev Dodgson said:

JudgeMental said:

SCIENCE said:

dv said:

The Rev Dodgson said:

It’s actually not that easy. In fact the 3 lowest integers that satisfy the equation are 79, 80 and 81 digits long.

Sorry to disappoint my adoring fans but this has me stymied.

I’m leaning towards saying there’s no such solution but then we are left with a need to prove that.

It is kind of a scalable problem. Like if you have a solution a,b,c then ka, kb, kc is a solution. Therefore if there is a solution, there is a solution in which a b and c do not share a common factor.

ellipsis

speaking of keeping up we think the point from The Rev Dodgson here is that Diophantine equation are fkn painful to solve

not for the ones I post. they let the participants fight it out. the one the rev posted he just made up or got it from somewhere.

Detailed answer

Not so detailed answer

It’s from Quora :)

ah yes that is entirely possibly where we seed it

from link

I know this looks like random voodoo, but please believe me that it’s not. Once you have those transformations, a tedious but straightforward algebraic calculation confirms that

https://en.wikipedia.org/wiki/Birational_geometry

Now, as soon as you have a rational point on an elliptic curve, such as P=(−100,260) on our curve (2), you can start generating others using the chord and tangent technique, which we covered on a previous Quora episode.

a=154476802108746166441951315019919837485664325669565431700026634898253202035277999, b=36875131794129999827197811565225474825492979968971970996283137471637224634055579, c=4373612677928697257861252602371390152816537558161613618621437993378423467772036

This is a striking example of the way diophantine equations with tiny coefficients can have enormous solutions. This isn’t merely awe-inspiring, it is profound. The negative solution of Hilbert’s 10th problem means that the growth of the solutions as the coefficients get larger is an uncomputable function, for if it were computable, we would have had a simple algorithm for solving diophantine equations, and there isn’t one (simple or complex). Here, the correspondence 4→ 80-digit numbers, 178→ hundreds-of-millions-digit numbers and 896→ trillions of digits gives us a glimpse into the first few tiny steps of that monstrous, uncomputable function. Tweak the numbers in your equation, and the solutions promptly exceed anything that fits in our puny little universe.

(…)

Reply Quote

Date: 23/03/2023 09:23:27
From: SCIENCE
ID: 2011308
Subject: re: Discrete Mathematics

The Rev Dodgson said:

SCIENCE said:

The Rev Dodgson said:

The Rev Dodgson said:

JudgeMental said:

The Rev Dodgson said:

The Rev Dodgson said:

SCIENCE said:

Kothos said:

dv said:

SCIENCE said:

The Rev Dodgson said:

dv said:

The Rev Dodgson said:

dv said:

Kothos said:

dv said:

The Rev Dodgson said:

dv said:

I’m bound to say, that’s a very interesting and counterintuitive result. Cheers for that, Rev.

That’s what I thought :)

On the bright side, the algebra puzzle made me dig up my python software for doing arithmetic on arbitrarily large integers.

Nice.

You’re probably wondering what the prime factors of those numbers are.

c =2^2×17×41×109×1117×12373×26897614153×110716137702073×894909542439071×390741051529463416523003987

I’ll leave the rest as an exercise.

Any hints on how to factorise big numbers like that?

You’d have to ask wolfram alpha…

Ah.

Might have a browse in Python.

well you know

if there were an efficient way to factorise large numbers

we mean

LOL

I mean we’ve all got a copy of Numerical Methods but it’s pretty clear wolfram alpha have some kind of sauce that I don’t have since they can factorise an 80 digit number in seconds rather than weeks…

Half of cryptography is built around the idea that this is difficult.

yeah honestly we thought The Rev Dodgson was doing a sneaky and trying to get us to do his quantum computing homework for him

The Internet has kindly given me some Python code, so I’m going to have a go.

I may be some time.

Well it works for 29 digits, in about 30 seconds.

So how long for 80 digits?

well, that’s a whole new order of magnitude bigger type problem.

OK, but am I looking at a few days, millions of years, or a few billion lifetimes of the universe?

I guess the important number is the number of digits in the largest factor, which is 27 for the 79 digit number, and 15 for the 29 digit number.

For an easier task I thought I’d check if Wolfram Alpha actually got the right answer, or whether it went all ChatGPT and just made something up.

I’m sure you’ll all be pleased to know that
2**2*17*41*109*1117*12373*26897614153*110716137702073*894909542439071*390741051529463416523003987

does indeed =

‘4373612677928697257861252602371390152816537558161613618621437993378423467772036

Also the factors of
11193123069125255733930404403495413078162092382549228
are

which took less than 20 seconds.

wait we had to do a double take on that but yes it really does show

so yeah we’ve been around a while but we’ve never seen a natural number with no factors

hint: the [&#91;square bracket&#93;] entities are a thing

Call me a lazy bugger if you will, but I’ll just leave off the square brackets.
The factors are:
4, 17, 41, 109, 1117, 12373, 26897614153, 110716137702073, 894909542439071

(…)

Reply Quote

Date: 23/03/2023 09:23:47
From: SCIENCE
ID: 2011309
Subject: re: Discrete Mathematics

Ian said:

btm said:

dv said:

btm said:

The Rev Dodgson said:

btm said:

The Rev Dodgson said:

btm said:

The Rev Dodgson said:

btm said:

Maybe your software can tell me the prime factors of 2159634636951022080954795988362711072142859201593196909736448075115982428195824688894947671474649787

That’s a composite number, but it’s got 101 digits.

It’s thinking.

Do you know how many digits in the largest factor?

Yes.

Good

How many?

50.

In that case I doubt it will finish before the equinox.

Wolfram alpha didn’t find any factors in “standard time”. It offered to extend the search if I paid for it..

I’m tossing up whether I’m the kind of person who would be nerdsniped into paying 10 bucks to look at prime factors.

JFTR,

42794985769661716377051160916445832815555430391701 × 50464665383345761322625509494017477207325958511887 = 2159634636951022080954795988362711072142859201593196909736448075115982428195824688894947671474649787

Thanks. I’ll use that.

(…)

Reply Quote

Date: 23/03/2023 09:24:37
From: SCIENCE
ID: 2011310
Subject: re: Discrete Mathematics

dv said:

btm said:

dv said:

SCIENCE said:

JudgeMental said:

dv said:

SCIENCE said:

JudgeMental said:

dv said:

There are algorithms that can pretty quickly generate, for instance, the 12 trillionth prime number.

But the largest known prime is around 1.4 × 10^24,862,048, which is a stupid big number.

graham might disagree.

the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding

NHOH

graham, ronald graham. not to be confused with benjamin graham.

how about Alexander Graham Bell, imagine not hearing

Yep.

https://t5k.org/nthprime/algorithm.php

Thanks dv.

Note… you’ll seriously run out of steam or time when you try to use it as a means to work out, for instance, the 10^20th prime, even though we know much bigger primes than that.

But it’s a workable method of finding particular ordinal primes up to around a quadrillion or so. To be honest I don’t know what is the highest ordinary prime ever discovered by these or similar methods.

(…)

Reply Quote

Date: 23/03/2023 09:26:16
From: SCIENCE
ID: 2011312
Subject: re: Discrete Mathematics

SCIENCE said:

The Rev Dodgson said:

I was going to post some Python code here for the function I was using to factorise large numbers, but I can’t work out how to post code in a readable format, so I’ll just post some results:

The code is pretty simple, let me know if you’d like a link.

here let us bootstrap some of that for you, our next 2 posts in this thread will be alternative methods of delivering a HTML contained JS powered utility for posting code in a readable format

simply paste our code into a file and open it as HTML in your JS enabled browser (easiest if you give it a HTML type extension when saved), and you too will have the power

now for the code

Reply Quote

Date: 23/03/2023 09:27:00
From: SCIENCE
ID: 2011313
Subject: re: Discrete Mathematics
<!DOCTYPE HTML><html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>code for https://tokyo3.org/forums/holiday/</title></head><body>
<h1>code for <a href="https://tokyo3.org/forums/holiday/">https://tokyo3.org/forums/holiday/</a></h1>
<div>
<textarea id="i" cols="80" rows="25">paste code here to cure the whitespace problem, then select the output from the other box to copy and paste it to forum</textarea>
<textarea id="o" cols="80" rows="25"></textarea>
</div>
<div id="f">
</div>
<script type="text/javascript"><!--
"use strict";
(function () {
 document.getElementById("i").focus();
 document.getElementById("o").addEventListener("focus", function () {
  var i, n, o = "";
  i = document.getElementById("i").value;
  for (n = 0; n < i.length; n++) {
   switch (i[n]) {
    case " ": case "\xA0": {
     o += "&nbsp;";
    } break;
    case "\r": case "\n": {
     o += "<br>\n";
    } break;
    default: {
     o += "&#x" + i.charCodeAt(n).toString(16) + ";";
    }
   }
  }
  o = o.replace(/(\r|\n)/g, "");
  document.getElementById("f").innerHTML = document.getElementById("o").value = "<div style=\"font-family: Consolas, Lucida Console, Courier New, monospace;\">\n" + o + "\n</div>\n";
  document.getElementById("o").select();
 }, false);
})();
//--></script>
</body></html>
Reply Quote

Date: 23/03/2023 09:29:03
From: SCIENCE
ID: 2011314
Subject: re: Discrete Mathematics
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Reply Quote

Date: 23/03/2023 09:32:26
From: SCIENCE
ID: 2011315
Subject: re: Discrete Mathematics

dv said:

The Rev Dodgson said:

Peak Warming Man said:

The Rev Dodgson said:

Peak Warming Man said:

a = ninety seven
b = forty six
c = one thousand and three

Off by a bit over 3.
Approx 3.14 in fact.

No worries.

Well that was much easier than it looked like it was going to be :)

Thanks for your help. Well deserving of its own thread, if you feel so inclined.

Well you’re both fine fellows

(from before)

Reply Quote

Date: 23/03/2023 09:33:44
From: SCIENCE
ID: 2011316
Subject: re: Discrete Mathematics

The Rev Dodgson said:

Code pasted with the aid of SCIENCE:

@xl_func
@xl_arg('n', 'str')
@xl_return('str')
def factorization(n, rtntime = 0):
    n = int(n)
    factors = []
    if rtntime > 0: stime = time.perf_counter()
    def get_factor(n):
        x_fixed = 2
        cycle_size = 2
        x = 2
        factor = 1

        while factor == 1:
            for count in range(cycle_size):
                if factor > 1: break
                x = (x * x + 1) % n
                factor = gcd(x - x_fixed, n)

            cycle_size *= 2
            x_fixed = x

        return factor

    while n > 1:
        next = get_factor(n)
        factors.append(next)
        n //= next
    if rtntime > 0: factors.append(time.perf_counter()-stime)
    return factors

@xl_func
@xl_arg('n1', 'str')
@xl_arg('n2', 'str')
@xl_return('str')
def py_gcd(n1, n2):
    n1 = int(n1)
    n2 = int(n2)
    return gcd(n1, n2)

@xl_func
@xl_arg('n', 'str')
@xl_arg('div', 'str')
@xl_return('str')
def py_mod(n, div):
    n = int(n)
    div = int(div)
    return n % div

@xl_func
@xl_arg('n', 'str')
@xl_arg('div', 'str')
@xl_return('str')
def py_floor(n, div):
    n = int(n)
    div = int(div)
    return n // div

as cited, The Rev Dodgson provided this software

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Date: 23/03/2023 09:33:49
From: roughbarked
ID: 2011317
Subject: re: Discrete Mathematics

SCIENCE said:

dv said:

The Rev Dodgson said:

Well that was much easier than it looked like it was going to be :)

Thanks for your help. Well deserving of its own thread, if you feel so inclined.

Well you’re both fine fellows

(from before)

Time travel.

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Date: 23/03/2023 09:36:51
From: SCIENCE
ID: 2011320
Subject: re: Discrete Mathematics

For the convenience of everyone here, we have uploaded the pages that help to post code in a readable format.

http://void.byethost10.com/sssf/ForUmCodeEnt.html
http://void.byethost10.com/sssf/ForUmCodeB64.html

In future revisions and at request we may have it wrap base64 lines to 80 (or a user-defined number of) characters.

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Date: 23/03/2023 13:04:31
From: mollwollfumble
ID: 2011453
Subject: re: Discrete Mathematics

As always, the first step is to draw a graph.

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Date: 23/03/2023 18:53:29
From: SCIENCE
ID: 2011536
Subject: re: Discrete Mathematics

JudgeMental said:

SCIENCE said:

SCIENCE said:

dv said:

That’s not what writing means. Writing According to Hoyle has to represent a natural language fully.

seems unfair, next they’ll tell you that mathematics has to be complete and consistent, and then you realise there is no such thing as mathematics

… and then you realise there is no such thing as mathematics

Phew, nice to know I was useless at something that doesn’t exist.

Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; ie, there are statements of the language of F which can neither be proved nor disproved in F. For any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in F itself.” — Gödel via Raatikainen 2020

https://plato.stanford.edu/entries/goedel-incompleteness/
https://www.theguardian.com/science/2022/jan/10/can-you-solve-it-godels-incompleteness-theorem
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

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Date: 23/03/2023 18:53:53
From: SCIENCE
ID: 2011537
Subject: re: Discrete Mathematics

sorry, please remove excess quotation mark there

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Date: 23/03/2023 22:24:16
From: The Rev Dodgson
ID: 2011576
Subject: re: Discrete Mathematics

A little background on the code below:
It is Python code written so the functions can be called from Excel, using the pyxll add-in. The lines starting with @xl_ are to indicate that the function is to be callable from Excel, and to define the data types of the arguments and return values. Since Excel can’t handle long numbers everything is passed as strings, then converted to integers in the Python code.

Other than that, all the code should work from any Python interface. Just delete all the @xl_ decorators, and the n = int(n) lines.

If you want to run it from Excel but don’t want to pay for pyxll you can use xlwings, which does much the same thing.

SCIENCE said:

The Rev Dodgson said:

Code pasted with the aid of SCIENCE:

@xl_func
@xl_arg('n', 'str')
@xl_return('str')
def factorization(n, rtntime = 0):
    n = int(n)
    factors = []
    if rtntime > 0: stime = time.perf_counter()
    def get_factor(n):
        x_fixed = 2
        cycle_size = 2
        x = 2
        factor = 1

        while factor == 1:
            for count in range(cycle_size):
                if factor > 1: break
                x = (x * x + 1) % n
                factor = gcd(x - x_fixed, n)

            cycle_size *= 2
            x_fixed = x

        return factor

    while n > 1:
        next = get_factor(n)
        factors.append(next)
        n //= next
    if rtntime > 0: factors.append(time.perf_counter()-stime)
    return factors

@xl_func
@xl_arg('n1', 'str')
@xl_arg('n2', 'str')
@xl_return('str')
def py_gcd(n1, n2):
    n1 = int(n1)
    n2 = int(n2)
    return gcd(n1, n2)

@xl_func
@xl_arg('n', 'str')
@xl_arg('div', 'str')
@xl_return('str')
def py_mod(n, div):
    n = int(n)
    div = int(div)
    return n % div

@xl_func
@xl_arg('n', 'str')
@xl_arg('div', 'str')
@xl_return('str')
def py_floor(n, div):
    n = int(n)
    div = int(div)
    return n // div

as cited, The Rev Dodgson provided this software

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Date: 8/04/2023 15:17:47
From: SCIENCE
ID: 2017014
Subject: re: Discrete Mathematics

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Date: 8/04/2023 15:34:23
From: Michael V
ID: 2017016
Subject: re: Discrete Mathematics

SCIENCE said:


Nice tessellation.

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Date: 8/04/2023 15:39:42
From: SCIENCE
ID: 2017017
Subject: re: Discrete Mathematics

Michael V said:

SCIENCE said:


Nice tessellation.

https://cs.uwaterloo.ca/~csk/hat/

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, 2023

An aperiodic monotile, sometimes called an “einstein”, is a shape that tiles the plane, but never periodically. In this paper we present the first true aperiodic monotile, a shape that forces aperiodicity through geometry alone, with no additional constraints applied via matching conditions. We prove that this shape, a polykite that we call “the hat”, must assemble into tilings based on a substitution system. The drawing above shows a patch of hats produced using a few rounds of substitution.

https://arxiv.org/abs/2303.10798

Penrose got aperiodic tessellation down to 2 tiles like 50 years ago and this kind of stuff looks like child’s play so it’s quite something that it’s taken that long to get confirmation of 1 tile.

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Date: 8/04/2023 15:50:16
From: The Rev Dodgson
ID: 2017019
Subject: re: Discrete Mathematics

SCIENCE said:


Michael V said:

SCIENCE said:


Nice tessellation.

https://cs.uwaterloo.ca/~csk/hat/

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, 2023

An aperiodic monotile, sometimes called an “einstein”, is a shape that tiles the plane, but never periodically. In this paper we present the first true aperiodic monotile, a shape that forces aperiodicity through geometry alone, with no additional constraints applied via matching conditions. We prove that this shape, a polykite that we call “the hat”, must assemble into tilings based on a substitution system. The drawing above shows a patch of hats produced using a few rounds of substitution.

https://arxiv.org/abs/2303.10798

Penrose got aperiodic tessellation down to 2 tiles like 50 years ago and this kind of stuff looks like child’s play so it’s quite something that it’s taken that long to get confirmation of 1 tile.

Was reading about that in my New Scientist this morning.

I don’t really get this whole aperiodic thing I must confess.

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Date: 8/04/2023 15:57:51
From: SCIENCE
ID: 2017022
Subject: re: Discrete Mathematics

The Rev Dodgson said:

SCIENCE said:

Michael V said:

Nice tessellation.

https://cs.uwaterloo.ca/~csk/hat/

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, 2023

An aperiodic monotile, sometimes called an “einstein”, is a shape that tiles the plane, but never periodically. In this paper we present the first true aperiodic monotile, a shape that forces aperiodicity through geometry alone, with no additional constraints applied via matching conditions. We prove that this shape, a polykite that we call “the hat”, must assemble into tilings based on a substitution system. The drawing above shows a patch of hats produced using a few rounds of substitution.

https://arxiv.org/abs/2303.10798

Penrose got aperiodic tessellation down to 2 tiles like 50 years ago and this kind of stuff looks like child’s play so it’s quite something that it’s taken that long to get confirmation of 1 tile.

Was reading about that in my New Scientist this morning.

I don’t really get this whole aperiodic thing I must confess.

Well all right, we might have to rmit, neither do we.

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Date: 2/05/2023 22:39:17
From: SCIENCE
ID: 2026543
Subject: re: Discrete Mathematics

The Rev Dodgson said:

SCIENCE said:

The Rev Dodgson said:

My learnin for today:

Ramanujan wrote that:

pi =(approx) (9^2 + 19^2/22)^¼

which can also be written:

(3^7/22-2)^¼

and these are accurate to 8 decimal places, but being an engineer, I can’t resist adding 0.000000001, which makes it accurate to 10 decimal places.

So pi = (3^7/22-2)^¼ + 1e-9

Exactly

near enough for any reasonable person.

We’re happy with 10^0 for now.

You’re saying that the shortest distance between two points may be a straight line or a semi-circle?

Depends on which P you use when you norm L^P we guess.

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Date: 2/08/2023 20:07:50
From: SCIENCE
ID: 2060861
Subject: re: Discrete Mathematics

sarahs mum said:

buffy said:

OCDC said:

buffy said:

OCDC said:

buffy said:

OCDC said:
Dinner report: roasted skin on chicken thigh sprinkled with winter soup mix herb blend, to be served with broccoli and gravy

We are having fish and chips and steamed Romanesco broccoli. Mr buffy and my Melbourne brother have gone around the corner to get the fish and chips.

buffy, buffy, buffy. One does not have fresh veg with F&C.

But I found a beautiful Romanesco in the garden and I love fractal veggies.

:)

What other fractal vegies are there?

Dunno. Maybe cauliflower.

do sunflowers count?

Apparently they do, in a recurrence relation of s(n) = s(n-1) + s(n-2).

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Date: 2/08/2023 20:44:38
From: SCIENCE
ID: 2060863
Subject: re: Discrete Mathematics

Ah so the nerdsniping was effective.

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Date: 5/11/2023 18:22:11
From: SCIENCE
ID: 2091443
Subject: re: Discrete Mathematics

So we have to admit, we didn’t know this.

There are two common definitions of the trapezium. The American definition is a quadrilateral with no parallel sides; the British definition is a quadrilateral with two sides parallel (e.g., Bronshtein and Semendyayev 1977, p. 174)—which Americans call a trapezoid.

Definitions for trapezoid and trapezium have caused controversy for more than two thousand years.

fuck

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Date: 5/11/2023 18:31:09
From: Bubblecar
ID: 2091451
Subject: re: Discrete Mathematics

SCIENCE said:

So we have to admit, we didn’t know this.

There are two common definitions of the trapezium. The American definition is a quadrilateral with no parallel sides; the British definition is a quadrilateral with two sides parallel (e.g., Bronshtein and Semendyayev 1977, p. 174)—which Americans call a trapezoid.

Definitions for trapezoid and trapezium have caused controversy for more than two thousand years.

fuck

Also, in Australian recipes one tablespoon = four teaspoons. Most other places, one tablespoon = three teaspoons.

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Date: 22/10/2024 20:38:44
From: SCIENCE
ID: 2207696
Subject: re: Discrete Mathematics

Don’t Say We Never Give Yous Good News

https://www.mersenne.org/primes/?press=M136279841

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Date: 22/10/2024 20:43:54
From: Bubblecar
ID: 2207698
Subject: re: Discrete Mathematics

SCIENCE said:

Don’t Say We Never Give Yous Good News

https://www.mersenne.org/primes/?press=M136279841

Let’s hope they use it wisely.

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Date: 22/10/2024 20:48:52
From: SCIENCE
ID: 2207700
Subject: re: Discrete Mathematics

Bubblecar said:

SCIENCE said:

Don’t Say We Never Give Yous Good News

https://www.mersenne.org/primes/?press=M136279841

Let’s hope they use it wisely.

exactly, these nerds should have stuck to mining cryptocurrency, what’s another prime going to do for their wallet anyway

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