## Proportional Dice

So, here’s a nice probability problem that recently made it to my Twitter friends page:

(By the way, I’m @Nebusj on Twitter. I’m happy to pick up new conversational partners even if I never quite feel right starting to chat with someone.)

Schmidt does assume normal, ordinary, six-sided dice for this. You can work out the problem for four- or eight- or twenty- or whatever-sided dice, with most likely a different answer.

But given that, the problem hasn’t quite got an answer right away. Reasonable people could disagree about what it means to say “if you roll a die four times, what is the probability you create a correct proportion?” For example, do you have to put the die result in a particular order? Or can you take the four numbers you get and arrange them any way at all? This is important. If you have the numbers 1, 4, 2, and 2, then obviously 1/4 = 2/2 is false. But rearrange them to 1/2 = 2/4 and you have something true.

We can reason this out. We can work out how many ways there are to throw a die four times, and so how many different outcomes there are. Then we count the number of outcomes that give us a valid proportion. That count divided by the number of possible outcomes is the probability of a successful outcome. It’s getting a correct count of the desired outcomes that’s tricky.

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• #### howardat58 3:14 pm on Wednesday, 10 February, 2016 Permalink | Reply

Vegetarians clearly have different definitions.

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• #### Chiaroscuro 4:00 pm on Wednesday, 10 February, 2016 Permalink | Reply

So, let’s make these A/B=C/D for the dice, assuming in-order rolls. 1296 possibilities.

If A=C and B=D, it’ll always work. So that’s 36.

Additionally: If A=B and C=D, it’ll always work (1=1). So that’s 36,. minus the 6 where A=B=C=D.

Then 1/2=2/4 (and converse and inverse and both), 1/2=3/6 (same), 2/4=3/6 (same), 1/3=2/6 (same). 4, 4, 4, 4.so 16 total.

36+30+16=82, unless I’ve missed some. 82/1296, which reduces to 41/648.

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• #### Chiaroscuro 4:02 pm on Wednesday, 10 February, 2016 Permalink | Reply

Ooh! I missed 2/3=4/6. (and converse, and inverse, and both). So another 4, meaning 86/1296.

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## Reading the Comics, February 6, 2016: Lottery Edition

As mentioned, the lottery was a big thing a couple of weeks ago. So there were a couple of lottery-themed comics recently. Let me group them together. Comic strips tend to be anti-lottery. It’s as though people trying to make a living drawing comics for newspapers are skeptical of wild long-shot dreams.

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• #### fluffy 7:01 pm on Thursday, 28 January, 2016 Permalink | Reply

I didn’t know that was necessary. So I do like $a^3 + b^3 = c^3$?

A preview function would be nice.

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• #### Joseph Nebus 10:36 pm on Thursday, 28 January, 2016 Permalink | Reply

That’s the way, yes. WordPress’s commenting system certainly needs a preview function and an edit button.

(There might be other themes that have preview functions. The one I’m using here is a bit old-fashioned and it might predate a preview. I know it isn’t really mobile-friendly.)

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• #### Joseph Nebus 10:32 pm on Tuesday, 26 January, 2016 Permalink | Reply

It’s certainly possible. Starting from $\frac{N}{x} = x - N$ we get the equation $0 = x^2 - Nx - N$ and that has solutions $x = \frac{1}{2} \left( N \pm \sqrt{N^2 + 4N}\right)$.

I don’t seem able to include the table that would list the first couple of these without breaking the commenting system. But it’s easy to generate from that start. The x_N for a given N gets to be quite close to N+1.

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• #### Eric Mann 11:55 am on Wednesday, 27 January, 2016 Permalink | Reply

These x_N’s are nice little multiples of the (dare I say) golden ratio, yes? They will arise from a rectangle with dimensions of x and N with an embedded N by N square.

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• #### Joseph Nebus 10:35 pm on Thursday, 28 January, 2016 Permalink | Reply

I don’t see any link between these x_N’s or the golden ratio. Could you tell me what you see, please?

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• #### Eric Mann 11:51 am on Wednesday, 27 January, 2016 Permalink | Reply

Well done. I love the inquiry. What do the rectangles look like? Or, is there still a geometric interpretation?

I admit I still find the sequence of rectangles with Fibonacci dimensions and an embedded spiral attractive. I appreciate it in the limit. I am not attached to the ratio being golden with a capitol g, but is it just a curiosity? An attraction? I expect more out of the ratio, rectangle, and spiral.

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• #### Joseph Nebus 10:32 pm on Thursday, 28 January, 2016 Permalink | Reply

Well, rectangles in these gilt-ratio proportions would be longer and skinnier things. For example, you might see a rectangle that’s one inch wide and 20.049(etc) inches long. I doubt anyone could tell the difference between that and a rectangle that’s one inch wide, 20 inches long, though.

I do think it’s just a curiosity, an attractive-looking number. Or family of numbers, if you open up to these sorts of variations. There’s nothing wrong with looking at something that’s just attractive, though. It’s fun, for one thing. And the thinking done about one problem surely helps one practice for other problems. I was writing recently about the Collatz Conjecture. As far as I know nothing interesting depends on the conjecture being true or false, but it’s still enjoyable.

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## Reading the Comics, January 21, 2016: Andertoons Edition

It’s been a relatively sleepy week from Comic Strip Master Command. Fortunately, Mark Anderson is always there to save me.

In the Andertoons department for the 17th of January, Mark Anderson gives us a rounding joke. It amuses me and reminds me of the strip about rounding up the 196 cows to 200 (or whatever it was). But one of the commenters was right: 800 would be an even rounder number. If the teacher’s sharp he thought of that next.

Andertoons is back the 21st of January, with a clash-of-media-expectations style joke. Since there’s not much to say of that, I am drawn to wondering what the teacher was getting to with this diagram. The obvious-to-me thing to talk about two lines intersecting would be which sets of angles are equal to one another, and how to prove it. But to talk about that easily requires giving names to the diagram. Giving the intersection point the name Q is a good start, and P and R are good names for the lines. But without points on the lines identified, and named, it’s hard to talk about any of the four angles there. If the lesson isn’t about angles, if it’s just about the lines and their one point of intersection, then what’s being addressed? Of course other points, and labels, could be added later. But I’m curious if there’s an obvious and sensible lesson to be given just from this starting point. If you have one, write in and let me know, please.

Ted Shearer’s Quincy for the 19th of January (originally the 4th of November, 1976).

Ted Shearer’s Quincy for the 19th of January (originally the 4th of November, 1976) sees a loss of faith in the Law of Averages. We all sympathize. There are several different ways to state the Law of Averages. These different forms get at the same idea: on average, things are average. More, if we go through a stretch when things are not average, then, we shouldn’t expect that to continue. Things should be closer to average next time.

For example. Let’s suppose in a typical week Quincy’s teacher calls on him ten times, and he’s got a 50-50 chance of knowing the answer for each question. So normally he’s right five times. If he had a lousy week in which he knew the right answer just once, yes, that’s dismal-feeling. We can be confident that next week, though, he’s likely to put in a better performance.

That doesn’t mean he’s due for a good stretch, though. He’s as likely next week to get three questions right as he is to get eight right. Eight feels fantastic. But three is only a bit less dismal-feeling than one. The Gambler’s Fallacy, which is one of those things everyone wishes to believe in when they feel they’re due, is that eight right answers should be more likely than three. After all, that’ll make his two-week average closer to normal. But if Quincy’s as likely to get any question right or wrong, regardless of what came before, then he can’t be more likely to get eight right than to get three right. All we can say is he’s more likely to get three or eight right than he is to get one (or nine) right the next week. He’d better study.

(I don’t talk about this much, because it isn’t an art blog. But I would like folks to notice the line art, the shading, and the grey halftone screening. Shearer puts in some nicely expressive and active artwork for a joke that doesn’t need any setting whatsoever. I like a strip that’s pleasant to look at.)

Tom Toles’s Randolph Itch, 2 am for the 19th of January (a rerun from the 18th of April, 2000) has got almost no mathematical content. But it’s funny, so, here. The tag also mentions Max Planck, one of the founders of quantum mechanics. He developed the idea that there was a smallest possible change in energy as a way to make the mathematics of black-body radiation work out. A black-body is just what it sounds like: get something that absorbs all light cast on it, and shine light on it. The thing will heat up. This is expressed by radiating light back out into the world. And if it doesn’t give you that chill of wonder to consider that a perfectly black thing will glow, then I don’t think you’ve pondered that quite enough.

Mark Pett’s Mister Lowe for the 21st of January (a rerun from the 18th of January, 2001) is a kid-resisting-the-word-problem joke. It’s meant to be a joke about Quentin overthinking the situation until he gets the wrong answer. Were this not a standardized test, though, I’d agree with Quentin. The given answers suppose that Tommy and Suzie are always going to have the same number of apples. But is inferring that a fair thing to expect from the test-takers? Why couldn’t Suzie get four more apples and Tommy none?

Probably the assumption that Tommy and Suzie get the same number of apples was left out because Pett had to get the whole question in within one panel. And I may be overthinking it no less than Quentin is. I can’t help doing that. I do like that the confounding answers make sense: I can understand exactly why someone making a mistake would make those. Coming up with plausible wrong answers for a multiple-choice test is no less difficult in mathematics than it is in other fields. It might be harder. It takes effort to remember the ways a student might plausibly misunderstand what to do. Test-writing is no less a craft than is test-taking.

• #### tkflor 4:46 am on Saturday, 23 January, 2016 Permalink | Reply

About black body radiation – for most practical purposes, a black body at room temperature does not emit radiation in the visible range. (https://en.wikipedia.org/wiki/Black-body_radiation)
So, we won’t see “light” or “glow”.

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• #### Joseph Nebus 5:03 am on Sunday, 24 January, 2016 Permalink | Reply

This is true, and I should have been clear about that. It glows in the sense that if you could look at the right part of the spectrum something would be detectable. It’s nevertheless an amazing thought.

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• #### Barb Knowles 6:20 pm on Saturday, 23 January, 2016 Permalink | Reply

This cartoon is GREAT! I’m going to show it to my math colleagues..

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• #### Joseph Nebus 5:04 am on Sunday, 24 January, 2016 Permalink | Reply

Glad you liked! I hope you make good use of it.

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## Some More Mathematics Stuff To Read

And some more reasy reading, because, why not? First up is a new Twitter account from Chris Lusto (Lustomatical), a high school teacher with interest in Mathematical Twitter. He’s constructed the Math-Twitter-Blog-o-Sphere Bot, which retweets postings of mathematics blogs. They’re drawn from his blogroll, and a set of posts comes up a couple of times per day. (I believe he’s running the bot manually, in case it starts malfunctioning, for now.) It could be a useful way to find something interesting to read, or if you’ve got your own mathematics blog, a way to let other folks know you want to be found interesting.

Also possibly of interest is Gregory Taylor’s Any ~Qs comic strip blog. Taylor is a high school teacher and an amateur cartoonist. He’s chosen the difficult task of drawing a comic about “math equations as people”. It’s always hard to do a narrowly focused web comic. You can see Taylor working out the challenges of writing and drawing so that both story and teaching purposes are clear. I would imagine, for example, people to giggle at least at “tangent pants” even if they’re not sure what a domain restriction would have to do with anything, or even necessarily mean. But it is neat to see someone trying to go beyond anthropomorphized numerals in a web comic. And, after all, Math With Bad Drawings has got the hang of it.

Finally, an article published in Notices of the American Mathematical Society, and which I found by some reference now lost to me. The essay, “Knots in the Nursery:(Cats) Cradle Song of James Clerk Maxwell”, is by Professor Daniel S Silver. It’s about the origins of knot theory, and particularly of a poem composed by James Clerk Maxwell. Knot theory was pioneered in the late 19th century by Peter Guthrie Tait. Maxwell is the fellow behind Maxwell’s Equations, the description of how electricity and magnetism propagate and affect one another. Maxwell’s also renowned in statistical mechanics circles for explaining, among other things, how the rings of Saturn could work. And it turns out he could write nice bits of doggerel, with references Silver usefully decodes. It’s worth reading for the mathematical-history content.

• #### elkement (Elke Stangl) 1:55 pm on Friday, 22 January, 2016 Permalink | Reply

Your blog is really an awesome resource for all things math, no doubt!!

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• #### Joseph Nebus 5:02 am on Sunday, 24 January, 2016 Permalink | Reply

That’s awfully kind of you to say. I’ve really just been grabbing the occasional thing that comes across my desk and passing that along, though, part of the great chain of vaguely sourced references.

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• #### elkement (Elke Stangl) 8:48 am on Sunday, 24 January, 2016 Permalink | Reply

But ‘curating’ as they say today is an art, too, and after all you manage to make things accessible, e.g. by summarizing posts you reblog so neatly…. and manage to do so without much images!!

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• #### Joseph Nebus 10:12 pm on Tuesday, 26 January, 2016 Permalink | Reply

Well, thank you again. I do feel like if I’m pointing to or reblogging someone else’s work I should provide a bit of context and original writing. It’s too easy to just pass around a link and say “here’s a good link”, which I wouldn’t blame anyone for doubting.

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